Algebra II Final – Flashcards

Unlock all answers in this set

Unlock answers
question
Hans pays $205 in advance on his account at the athletic club. Each time he uses the club, $10 is deducted from the account. Write an equation that represents the value remaining in his account after x visits to the club. Find the value remaining in the account after 11 visits.
answer
V = 205 - 10x; $95
question
Solve the proportion (x?8)/8 = 4/5
answer
x=14 or 72/5
question
Solve for x. x-7=1
answer
x=8
question
Solve the equation or formula for the variable specified. B = 7a²b, for b.
answer
b = B/(7a²)
question
Which equation best describes the graph below? (Evernote)
answer
y = -2x + 5
question
solve for x. 3x - 3 = x - 9
answer
x = -3
question
Which inequality has the solution shown in the graph? (Evernote)
answer
m + 7 ? 9
question
What inequality describes the graph? (Evernote)
answer
x ? -5
question
For the data given, find the equation of the line of best fit. (Evernote)
answer
Incorrect: None of the listed answers are correct Correct: y = 0.63x + 1.90
question
Write a proportion that models the statement 10 is to 5 as 16 is to 8.
answer
10/5 = 16/8
question
Write an equation in slope-intercept form for a line that passes through the given pair of points. (4,-7), (-2,-10)
answer
y = 1/2x - 9 Solution Slope Formula: slope(m) = (y2 - y1) / (x2 - x1) Substitute and Solve m = (-10 +7) / (-2 - 4) m = (-3) / (-6) m = 1/2 or 0.5 Slope intercept form: y = mx + b Pick one of the two points (4,-7) or (-2,-10), substitute and solve -7 = 1/2(4) + b -7 = 2 + b -7 - 2 = b -9 = b Final Answer y = 1/2x - 9
question
Write an equation in slope-intercept form for a line that passes through the given pair of points. (-6,5), (-9,0)
answer
y = 5x/3 + 15
question
Write an equation in slope-intercept form of the line that passes through (- 5, 4) and is parallel to the graph of y = -4x + 1.
answer
y = -4x -16
question
When Spheres-R-Us ships bags of golf balls, each bag must be within 5 balls of 370. Identify the inequality which results in an acceptable number of golf balls in each bag.
answer
365 ? 375
question
Abbiville is a small town that has been steadily growing since 1960. Use the table below to create a linear equation that estimates Abbiville's population over time. What will the population be in 2014 if the growth remains constant? (Evernote)
answer
364 people
question
Which is the equation of the line with slope -4 and y-intercept 5?
answer
4x + y -5 = 0 y = -4x + 5
question
A server at a restaurant kept track of the number of requests for the daily special, and the time of day the request was made. The data is displayed below. (Evernote)
answer
Incorrect: It is a linear model with a positive correlation.
question
Write the equation of the line, in slope-intercept form, that contains the point (-1, - 5) and is perpendicular to the line -8x - 4y = 3.
answer
y = -2x -7
question
Write the slope-intercept form of an equation of the line that passes through the point (3, - 6) and has the slope m = -4.
answer
y = -4x + 6
question
If x = 78 when y = 130 and x varies directly as y, then find x when y = 190.
answer
x = 114 when y = 190 Formula: y = ky(solving for y) or x = ky(solving for x) Substitute 78 = k(130) Divide by 130 on both sides k = 0.6 x = 0.6(190) x = 114
question
Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
answer
move it 3 to the right
question
Evaluate the expression. (2/5)?
answer
32/3125
question
Which of the following is an example of the Commutative Property of Addition?
answer
5 + 2 = 2 + 5
question
Evaluate the expression. 64 × 4² - 2 × 2²
answer
1016
question
Ivan Bogdanovich plans to decorate stuffed animals to sell at a crafts fair. The decorations cost $44.00 and the stuffed animals cost $5.25 each. a. Write a function expressing the cost, C(x), of the project in terms of the number of stuffed animals decorated, x. b. Determine the cost of decorating 25 stuffed animals. c. How many stuffed animals can be decorated with a budget of $227.75?
answer
a. C(x) = 5.25x + 44.00 b. $175.25 c. 35
question
For the pair of functions, f and g, find (g o f)(x) and (f o g)(x). f(x) = 6 - x, g(x) = x² - 2
answer
(g o f)(x) = x² - 12x + 34 Solution (6-x)² - 2 36 - 6x - 6x + x² - 2 x² - 6x - 6x + 36 - 2 x² - 12x + 34 (f o g)(x) = -x² + 8 Solution -x(x² - 2) + 6 -(x² - 2) + 6 -x² + 2 + 6 -x² + 8
question
Evaluate the expression. (3/8)?²
answer
64/9
question
Describe how the graph of the function y = 15(x + 2) + 4 can be obtained from the graph of y = 3(x + 2).
answer
shift up 4 units, and vertically stretch by a factor of 5
question
Simplify the expression. ((-u)³(-u?)?)/(u?)³
answer
-u?? Solution (-u³ × u?²)/u¹? -u??/u¹? = -u??
question
Evaluate the expression. (-3/8)²
answer
9/64
question
The graph of f(x) = x² is stretched vertically by a factor of 3, translated 4 units to the right, and translated 8 units downward. Determine the equation of the transformed function, g(x).
answer
g(x) = 3(x - 4)² - 8
question
A T-shirt company estimates that the average cost per shirt can be approximated by the function A(x) = (3.75x + 200)/x, where x is the number of T-shirts made. Find the average cost per T-shirt when the company makes 10 shirts.
answer
$23.75
question
Find the inverse of the function and determine whether the inverse is a function. y = 7x²
answer
y = ±?(x/7), y is not a function.
question
Find the inverse of the function and determine whether the inverse is a function. f(x) = 19x²
answer
f?¹(x) = ±?(x/19), f?¹(x) is not a function.
question
Since 1993, Daphne Hamilton has owned a franchise of take-out restaurants called The Burger Barn. The number of customers, C, in thousands, that The Burger Barn has served each year can be modeled by the function C(t) = t² + 38t + 600, where t is the number of years from 1993. Using this model, estimate the number of customers served in 1999.
answer
864,000
question
If f(x) = 9 - x² and g(x) = 3 - x, which is the rule of function (f × g)(x)?
answer
x³ - 3x² - 9x + 27
question
Evaluate the expression. 3 - 14 × 6 ÷ 7 + 3
answer
-6
question
If f(x) = 3x + 4 and g(x) = 2 + x, which is the rule of function 3g(x) - 3f(x)?
answer
-6x - 6
question
Which property is shown by the following statement? (13 + 3)+ 8 = 13 + (3 + 8)
answer
The Associative Property of Addition
question
Given the graph of y = 3(x + 1), what function would be obtained by moving the graph up 4 and moving it 6 to the right?
answer
f(x) = 3(x - 5) + 4
question
Use elimination to solve the system of equations. {7x - 3y = -43 {5x + 6y = -47
answer
(-7, -2)
question
Write the pair of parametric equations as a single equation in x and y. {x = 8t - 4 {y = 6t² - 2
answer
y = 6((x+4)/8)² - 2 Solution x = 8t - 4 Add 4 to both sides x + 4 = 8t Divide by 8 on both sides t = (x + 4)/8 Substitute y = 6((x+4)/8)² - 2
question
Mr. Reaich, the ticket agent for a small commuter airline, has developed a model to forecast the total monthly revenue from the airline's passenger service. Under this model, the total revenue, R, is equal to $368 times the number of first-class, F, plus $324 times the number of coach passengers, C. Their flight schedule and overall aircraft capacity limit the total number of passengers carried during a single month to a maximum of 30,775. The number of coach class passengers is at least seven times the number of first-class passengers, and logically, the number of airline passengers in either class cannot be negative. Express the model in mathematical form.
answer
$368F + $324C = R F + C ? 30,775 C ? 7F F ? 0; C ? 0
question
Solve the system of equations by substitution. {3x + 4y = 32 {3x + y = 17
answer
(4, 5)
question
Use a graph to solve the system of equations. {x + y = -1 {y = 3x + 19
answer
(-5, 4)
question
Graph and classify the system of equations as independent, inconsistent, or dependent. If the system is independent, find the solution from the graph. {3x + 3y = 2 {3x - y = 8
answer
Independent (2, -2)
question
Use elimination to solve the system of equations. {2x - 5y = 1 {5x -2y = -4
answer
(-22/21, -13/21)
question
Use elimination to solve the system of equations. {-16x - 6y = 170 {8x + 3y = -85
answer
Incorrect: no solutions Correct: Infinitely Many Solutions
question
Use a graph to solve the system of equations. {x + y = -4 {y = 3x - 8
answer
(1, -5)
question
Which of the given ordered pairs are NOT solutions to the system of inequalities? {2x + 3y 2 (3, 1), (-3, -2), (5, -5), (-2, 4), (-4, 5)
answer
(-4, 5), (3, 1), (-2, 4)
question
Write the system of inequalities whose solution is graphed as the shaded region.
answer
Incorrect: y -2x + 5 Use mathway; Graph the intersection (and)
question
Use the graph of the feasible region for the set of constraints to find the maximum and minimum values of the objective function C = 3x + 5y. Constraints: {x + y ? 1 {4x - 2y ? 4 {3y ? 3x + 3
answer
The maximum value of C is 29 at (3, 4) and the minimum value is 3 at (1, 0) .
question
Determine which ordered pair (x, y) is a solution of 3x - y ? 15.
answer
(5, 4) Solution Substitute 3(5) - 4 ? 15 11 ? 15 True
question
Which inequality has (1, 7) as a solution?
answer
Incorrect: -2x - 5y ? -32 Solution Substitute -2(1) - 5(7) ? -32 -2 - 35 ? -32 -37 ? -32 False, not a solution of the inequality
question
Use elimination to solve the system of equations. {4x + 2y = -12 {3x - 2y = 5
answer
(-1, -4)
question
An airplane, flying at an altitude of 3.21 miles above the ground, has a horizontal speed of 350 miles per hour and is descending at a rate of 14 miles per hour. Use parametric equations to find the airplane's altitude above ground after it has traveled 11 miles, as measured along the ground.
answer
2.77 mi Solution x(t)=350t y(t)=3.21-14t Solve either equation for "t" x=350t t=x/350 Substitute y=3.21-14(x/350) Substitute for "x", x=11 y=3.21-14(11/350) y=2.77
question
Use the set of constraints to find the maximum and minimum values of the objective function C = 5x + 11y Constraints: {x + y ? 2 {6x - 6y ? 12 {8y ? 4x + 16
answer
Incorrect: The maximum value of C is 139 at (8, 9) and the minimum value is 10 at (0, 2). Correct: The maximum value of C is 106 at (8,6) and the minimum value is 10 at (2,0).
question
Ms. Kim, an analyst at Multi-Fastener Corp, has developed a model to forecast the company's total production of metal fasteners. Under this model, production, P, is equal to 4500 times the man-hours spent producing machine screws (S) plus 2900 times the man-hours spent producing nuts and bolts (P). Manpower constraints mean that total number of man-hours per year cannot exceed 44,000; logically, the number of man-hours assigned to either task cannot be negative. Express the model in mathematical form.
answer
P = 4500S + 2900B {S + B ? 44,000 {S ? 0 {B ? 0
question
Solve the system of equations by substitution. {4y = 4 {x + 2y + z = 10 {y - 2z = -5
answer
Incorrect: None of the listed answers are correct Correct: x=5 y=1 z=3
question
Write the pair of parametric equations as a single equation in x and y. {x = 8t² -3 [y = 6t - 7
answer
x = 8((y + 7)/6)² - 3
question
Find [1 17 14][11 3] [19 7] [9 5]
answer
[460 192]
question
Which system of equations can be classified as independent?
answer
none of these
question
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {4x + y = -24 {2x + 2y = -18
answer
(-5, -4) You can just solve by using Elimination or Substitution
question
Find the determinant, and tell whether the matrix has an inverse. det [10 -7] [-5 -6 ]
answer
-95; Yes
question
Write the augmented matrix for the system of equations. {x - y = -9 {5x = 0
answer
[1 -1| -9] [5 0| 0] Solution x=1, y=-1 5x=5
question
The band and the cheerleading squad at a local school are ordering supplies. The supplies they need are listed in the table. (Evernote)
answer
[10 13 5] [ 5 ] [10 16 8] [12] [ 2 ] = [216] [258]
question
Perform the indicated operations on the given matrices. [3 -6 -7] [-2 -1 7] [-3 -2 4] + [ 3 5 -7] [-4 -1 7] [4 -9 -3]
answer
[ 1 -7 0 ] [ 0 3 -3 ] [0 -10 4]
question
Write the augmented matrix for the system of equations. {8x + 2y + 6z = -3 {3x - y + 10z = 3 {-9x -2y -4z = -7
answer
[8 2 6| -3 ] [3 -1 10| 3 ] [-9 -2 -4 |-7 ]
question
The Student Government secretary, and two teacher's aides are ordering supplies. The supplies they need are listed in the table. (Evernote)
answer
(Evernote)
question
Kozinski played in 16 basketball games this season, Hussein played in 32 games, and Johnson played in 55 games. Kozinski averaged 6 points and 2 rebounds per game, Hussein averaged 8 points and 11 rebounds, and Johnson averaged 18 points and 16 rebounds. Multiply the following matrices to get the total number of points scored and the total number of rebounds made, by all three players combined. [ 6 2 ] [16 32 55] [ 8 11] [18 16]
answer
1342 points, 1264 rebounds
question
Solve: (Evernote) det[x+3 2] [3 x-2] =0
answer
Incorrect: x =-4 Correct: x = 0 Mathway: Find the Determinant
question
Classify the following system of equations as inconsistent, dependent, or independent: {-3x -7y -2z = -25 {-6x +21y -6z = 57 {-3x +14y -4z = 40
answer
independent, because there is an intersection
question
Find the reduced row-echelon form of the augmented matrix that represents the following system of equations: (Evernote) {-6x + 9y = -5 {-8x - 4y = 0
answer
Incorrect: [1 0| 5/24 ] [0 1|-5/12] Correct: [1 0| -5/24] [0 1| 5/12
question
Perform the indicated operations on the given matrices. (Evernote) [-8 -8] + [-2 3] [-6 4] + [3 -1] (2×2 matrice + 2×2 matrice)
answer
[-10 -5] [-3 3]
question
Find AC - CB (Evernote)
answer
[-13 19] [-12 -44]
question
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {3x + 2y + z = -2 {3x -2y -z = -22 {3x + 2y -z = -10
answer
[3 2 1] [x] [ -2 ] [3 -2 -1] [y] = [-22] [3 2 -1] [z] [-10] ;(-4, 3, 4)
question
Find the inverse of the following matrix (if it exists): [2 -13 -6] [0 5 0] [0 -24 0]
answer
The matrix has no inverse.
question
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {x + 2y + z = -17 {3x + 7y + 2z = -56 {x - y + 2z = 0
answer
x = -4, y = -6, z = -1
question
In May, Victor bought 24 styrofoam balls and decorated them as toy figurines. In June, he sold 17 figurines. In May, Wanda bought 28 styrofoam balls to decorate and in June, she sold 12 figurines. Which matrix represents all of their May purchases and their June sales?
answer
May June Victor [24 -17] Wanda [28 -12]
question
Which matrix represents the graph of polygon PQRS? (Evernote)
answer
P Q R S [2 -3 -3 2] [6 2 -7 3]
question
Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 4x² = 8
answer
±?2; ±1.41
question
Use factoring and the Zero-Product Property to solve the quadratic equation 2x² + 5x -3 = 0
answer
1/2, -3 Use Quadratic equation: (-b ± ?((b² - 4ac))/2a
question
Find a quadratic function that fits the set of data points exactly, in the form y = ax² + bx + c with values of a, b, and c to two decimal places. (1, 4.1), (5, 2.3), (11, 4.9)
answer
y = 0.09x² - 0.98x + 4.99 Mathway: Find the Equation of the Parabola
question
Which function is the correct match with the the given graph and vertex (Evernote)
answer
f(x) = -3(x + 1)²
question
Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. y = -6x² - 24x - 31
answer
Incorrect: y = 6(x + 2)² - 7 x = -2; (2,-7) Correct: y = -6(x + 2)² - 7 x = -2; (-2, -7) Mathway: Find the Vertex Form Find the Axis of Symmetry Find the Vertex (Coordinates)
question
Use the quadratic formula to solve the equation. 6x² - 24x - 30 = 0
answer
5, -1
question
Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 2(x + 4)² - 48 = 0
answer
Incorrect: 4 ± 2?6; 5.8 or -13.9 Correct: -4 ± 2?6; 0.9 or -8.9
question
Simplify: (2 + 5i)/(5 + 4i)
answer
(30/41) + (17/41)i
question
Solve the quadratic inequality, and graph the solution on a number line. x² - 2x ? 24
answer
x ? -4 or x ? 6
question
Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. y = x² + 14x + 4
answer
Incorrect: None of the listed answers are correct Correct: y = (x+7)² - 45 x = -7; (-7, -45)
question
Which inequality is best represented by the graph shown: (Evernote)
answer
y ? -x² + 5x
question
Use factoring and the Zero-Product Property to find the zeros of the quadratic function. f(x) = x² -7x - 8
answer
8, -1
question
Which of the following is a quadratic function?
answer
f(x) = -1 + 9x² + 7x
question
Find the value of x to the nearest hundredth.
answer
17.2
question
Solve the quadratic inequality, and graph the solution on a number line. (Evernote) x² - 4x + 3 ; 0
answer
1 ; x ; 3
question
Factor the quadratic expression. x² - 11x + 30
answer
(x - 5) (x - 6)
question
Gupta threw a baseball off a cliff into an open field 50 feet below. The chart gives the horizontal distance, x (in feet), the baseball traveled from Gupta and the height, y (in feet), of the baseball above the field. (Evernote) distance, x | 8 | 18 | 33 | 43 | height, y | 65 | 76 | 79 | 71 | Choose the quadratic equation that best fits the baseball's trajectory from Gupta to the open field below.
answer
y = -0.0383x² + 2.143x + 50
question
Show that the function is a quadratic function by writing it in the form f(x) = ax² + bx + c and indentifying a, b and c. f(x) = (5x - 4) (5x + 3)
answer
f(x) = 25x² - 5x - 12 a = 25, b =5, c = -12
question
The turnstiles at the entrance to the State Fair kept track of the number of people entering the fairgrounds, for the first seven hours following the openning of the fair. Find a quadratic equation that models the data shown. (Evernote)
answer
0.083x² - 0.197x + 7.62 Graphing Calculator STAT?Edit Put x values into L1 and y values into L2. 2ND?MODE?STAT?CALC ?5:QuadReg
question
Solve the equation by completing the square. Give exact solutions. x² - 2x - 24 = 0
answer
-4, 6
question
Solve the equation for x. Write the exact solution and the approximate solution to the nearest hundredth, when appropriate. In(8x - 6) = 3
answer
x = (3/4) + (e³/8) x ? 3.26 Exponential form e³= 8x - 6
question
Solve the equation for x: 10^x = 18
answer
1.26
question
Evaluate 8^log? 9
answer
9
question
Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: In 10
answer
2.303
question
Solve log? 8 - log? (x + 3) = log? 3, for x
answer
Incorrect: None of the listed answers are correct Correct: x=-1/3 Solution log(x) - log(y) = log(x/y) log? 8 - log? (x+3) = log? (8/(x+3)) log? (8/(x+3)) = log? 3 (Mathway can solve once the left terms are combined)
question
If a principal of $1250 is invested at an annual interest rate of 4% compounded annually, what is the account balance at the end of 6 years?
answer
$1581.648
question
Solve the equation 3^4x = 27^x+3
answer
x = 9
question
The magnitude of an earthquake is found by the equation M = 2/3 log E/10^11.8 , where M is the magnitude and E is the energy released. Find the magnitude of an earthquake that released 10^24.4 ergs of energy.
answer
Incorrect: None of the listed answers are correct Correct: M = 8.4 Solution M = 2/3 log?? (10^24.4/10^11.8)
question
The formula for estimating the number, N, of a certain product sold is N = 7400 In(7t + 3), where t is the number of years after the product is introduced. What is the expected number of sales 7 years after the product is introduced? Round to the nearest whole number.
answer
29,239
question
Solve the equation. Round your answers to the nearest hundredth. 5^x-1 = 25
answer
3
question
Find the final amount of the investment: $3000 at 8% interest compounded quarterly for 4 years.
answer
$4118.36
question
Describe the value of b for the function y(x) = 1/3(1/b)^x to represent exponential growth.
answer
Incorrect: None of the listed answers are correct Correct: "b" has to be greater than 0 and less than 1. (0 < b < 1)
question
Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: e^¼
answer
1.284
question
Solve log?(x+4) - log?(x-4) = log?5 for x
answer
6
question
The inflation rate of the U.S. dollar is 3.1 percent. What this means is that every year, prices increase by 3.1 percent. If a pound of meat cost $2.37 nine years ago, what does it cost now?
answer
$3.12 Solution 2.37(1+0.031)?
question
Erosion gradually reduces the size of a small Pacific island that has a current area of just 460 acres. If the island's area decreases at an annual rate of 0.05%. Find the multiplier for the rate of exponential decay.
answer
Incorrect: 0.95 Correct: 0.9995 Solution 100 - 0.05 = 99.95 99.95/100 = 0.9995
question
Write the equation in logarithmic form. 6^-4 = 1/1296
answer
log? 1/1296 = -4
question
Which function represents exponential decay?
answer
y(x) = 5(0.44)^x
question
Evaluate the logarithmic expression to the nearest thousandth. log? 1/2
answer
-0.333
question
If $1500 is invested at an interest rate of 10%, compounded continuously, determine the balance in the account after 2 years. Use the formula A = Pe^rt.
answer
$1832.10
question
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x? - 9x³ + 10x² + 90x - 200
answer
4, 5, ±3.16
question
Divide x³ + 5x² + 9x + 9 ÷ x + 3
answer
x² + 2x + 3
question
Use a graph, synthetic division, substitution, and factoring to solve the equation. x³ - 7x² + 14x - 8 = 0
answer
2, 1, 4
question
Find all the zeros of the polynomial function. f(x) = x² - 10x + 34
answer
Incorrect: 5 ± 6i Correct: 5 ± 3i
question
The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1987. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 1998. (Evernote)
answer
T(x) = 0.2x³ + 0.3x² + 0.2x + 0.1; 304.8 thousand
question
Use a graph, synthetic division, substitution, and factoring to solve the equation. x³ - x² - 14x + 24 = 0
answer
-4, 2, 3
question
Find all the zeros of the polynomial function. f(x) = x³ + 4x² + 6x + 9
answer
-3, (-1 ±i?11)/2
question
Write the product as a polynomial in standard form. (5x² - 2x + 5)(x² - 8x - 2)
answer
5x? - 42x³ + 11x² - 36x - 10
question
Write a polynomial equation in standard form by using the given information. P is of degree 3; P(0) = 24; zeros: 3, -4, 2
answer
P(x) = x³ - x² - 14x + 24
question
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x³ - 4x² + 7x - 12
answer
None of the listed answers are correct
question
Simplify 3x² - 4x - 5 + 4x² + 8x - 7
answer
7x² + 4x - 12
question
Classify the polynomial by degree and number of terms. Describe the shape of its graph. 3x³
answer
cubic monomial; 'S' shaped with 2 turns
question
Evaluate 2x² + 3x + 4 when x=2
answer
18
question
For the function, use synthetic division and substitution to determine whether the given value is a zero of the function. P(x) = 3x? - 10x³ - 41x² + 68x + 60
answer
Zeroes are 5, 2, -2/3, -3
question
Evaluate x? - 10x² + 24 when x=3
answer
15
question
Write the product as a polynomial in standard form. (x - 6)(x + 5)(x - 4)
answer
x³ - 5x² - 26x + 120
question
Use substitution to determine which of the given linear expressions is a not a factor of 3x? - 16x³ - 15x² + 88x + 60
answer
x + 5 Mathway (Algebra) Factor
question
Determine the end behavior of the graph of the function f(x) = 3x? - 2x³ + 2x + 2
answer
rises to the left; rises to the right
question
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x? - x³ - 16x² + 10x + 60
answer
3, -2, ±3.16
question
Classify the polynomial by degree and number of terms. Describe the shape of its graph. 7x² + 7x + 7
answer
quadratic trinomial; parabola
question
Determine whether the function is a rational function. If so, find the domain and identify the horizontal and vertical asymptotes, and any holes in the graph. If the function is not rational, state why not. f(x) = (x - 1)/((x - 7)(x + 3))
answer
Incorrect: rational; x?1, 7 or -3; asymptotes at y=0, x=1, x=7 and x=-3 Correct: rational; x?-3,7 asymptotes at y=0, x=-3, and x=7
question
Simplify the rational expression. (Evernote) ( 3/(x² + 7x + 12) + 1/(x² + 11x + 28) ) ÷ ( 2/(x² + 10x + 21) + 1/x² + 13x + 36) )
answer
( 4(x + 9)(x + 6) ) / ( 3(x² + 12x + 31) )
question
Simplify. ( (x - 2)/(2x²) ) - ( (2x + 1)/(9x) ) + ( (7x)/(12) )
answer
( 21x³ - 8x² + 14x - 36 ) / 36x²
question
Solve the radical inequality. ³?(3x-6) + 6 ? 7
answer
x ? 7/3
question
Simplify the rational expression. ( (x² + 14x + 49)/8x ) ÷ ( (x+7)/4x )
answer
(x+7)/2
question
Evaluate the radical expression: (-1/7)(³?-64)?
answer
-256/7
question
Rationalize the denominator. 2 ÷ ?5
answer
(2?5)/5
question
Rationalize the denominator. (?11) ÷ (9 + ?5)
answer
( 9?11 - ?55) / 76
question
Simplify. (-3x + 8)/(x² - 49) - (-2x + 1)/(x² - 49)
answer
-1/(x+7)
question
Use the information to write the appropriate variation equation, and find y for the given values. y varies jointly as x and the inverse of z. y=-99.4 when x=-11 and z=4. Find y when x=2 and z=-8.
answer
y = 9x/z; -9/4
question
Simplify the sum, difference, product, or quotient. Assume that the value of any variable is positive ³?(-27x) - 4 ³?(x?) + 9 ³?(x) + 9x ³?(x)
answer
6 ³?(x) + 5x ³?(x)
question
Evaluate the radical expression: 8(??(81)²
answer
72
question
Solve the equation or inequality. 4/(x-3) - 2/(x+4) = 0
answer
x=-11
question
Find the domain of the radical function. f(x) - ?(9-x)
answer
x ? 9 Solution f(x) - ?(9-x) = 0 Add ?(9-x) to both sides f(x) = ?(9-x) Mathway: Find the Domain
question
Use the information to write the appropriate variation equation, and find y for the given values. y varies jointly as x and z. y=98/3 when x=2 and z=7. Find y when x=8 and z=3
answer
y = 7/3(x)(z); 56
question
Designer Dolls, Inc. found that the number of dolls sold, N, varies directly as their advertising budget, A, and inversely as the price of each doll, P. Designer Dolls, Inc. sold 8400 dolls when $84,000 was spent on advertising and the price of a doll was set at $70. Determine the number of dolls sold when the amount spent on advertising is increased to $98,000.
answer
9800
question
Write an equation that can be used to solve the problem. Then answer the question asked. A group of college students are volunteering for Habitat for Humanity during their spring break. They are putting the finishing touches on a house they built. Working alone, Dale Horton can paint a certain room in 3 hours. Kathy Garcia can paint the same room in 9 hours. How long will it take them working together to paint the room?
answer
x/3 + x/9 = 1; 2.25 hr
question
The wattage rating of an appliance is given as "W", in watts, and varies jointly as the resistance, "R", in ohms, and as the square of the current, "I", in amperes. If the wattage is 2 watts when the resistance is 200 ohms and the current is 0.1 amperes, find the wattage when the resistance is 150 ohms and the current is 0.4 amperes.
answer
Incorrect: None of the listed answers are correct Correct: W (wattage) = 24
question
Solve the equation or inequality. (2x-2)/(x²-36) ? 1/(x+6)
answer
-6 ? x ? -4 or x ? 6
question
Solve the radical inequality. ?(x + 14) ? x - 16
answer
x ? 22
question
Find the standard equation for the hyperbola with the given characteristics. vertices: (2, 0) and (-2, 0) asymptote: y=1/2x
answer
x²/4 - y² = 1 Put in each answer using Mathway: Find the Vertex Form, till the verices and asymptote matches up
question
Which equation describes a parabola?
answer
6y² + 11x - 13y = -1
question
Find the center and the radius of the circle that has a diameter with the given endpoints. Diameter CD, C(-1, -4), D(5, 4)
answer
center: (2, 0) radius: 5
question
Find the standard equation for the ellipse, using the given characteristics taken from the graph. (Evernote)
answer
(x+4)²/16 + (y+3)²/9 = 1
question
Write the equation in standard form and classify the conic section it defines. 2x² + 2y² - 28x + 24y + 162 = 0
answer
(x-7)² + (y+6)² = 4; circle
question
Find the center and the radius of the circle that has a diameter with the given endpoints. Diameter CD, with endpoints C(- 1, 5), D(5, 9)
answer
center: (2, 7) radius: ?13; 3.61 Mathway: Find the Midpoint
question
Find the standard equation of a circle with the given radius and center. radius: 7 center: (1, 1)
answer
(x - 1)² + (y - 1)² = 49
question
Solve the nonlinear system of equations. {x² + y² = 144 {x² - 4y² = 64
answer
four solutions: (±8?2, ±4)
question
Which set of characteristics matches the given standard equation and parabola graph? (Evernote) x = 1/24 y²
answer
vertex: (0, 0) directrix: x=-6
question
Solve the nonlinear system of equations. {x² + y² = 256 {x + y = 16
answer
two solutions: (0, 16) and (16, 0)
question
Which set of characteristics matches the given parabola graph and standard equation? (Evernote) x = -1/4 y²
answer
axis of symmetry: y=0 focus: (-1, 0) vertex: (0, 0) Mathway: Graph
question
Find the standard equation for the hyperbola with the given characteristics. center: (5, 4) one focus: (-25, 4) one vertex: (23, 4)
answer
(x-5)²/324 - (y-4)²/576 = 1
question
The two hyperbolas in the graph are conjugates, meaning they share the same asymptotes. Given the following equation of the hyperbola represented by the thin curve, find the equation of its conjugate, which is represented by the thick curve. (Evernote) x²/9 - y²/25 = 1
answer
y²/25 - x²/9 = 1
question
Identify the conic section produced by the following diagram. (Evernote)
answer
hyperbola
question
Find the standard equation for the hyperbola with the given characteristics. vertices: (2, 0) and (?2, 0) asymptote: y = 3x
answer
x²/4 - y²/36 = 1
question
Which set of characteristics matches the given standard equation and parabola graph? (Evernote)
answer
focus: (-1, -4) vertex: (-1, 2)
question
Identify the conic section produced by the intersection of the plane and the cones in the following diagram.
answer
circle
question
Find the standard equation of a circle with the given radius and center. radius: 4 center: (3, - 5)
answer
(x - 3)² + (y + 5)² = 16 Mathway: Find the Equation (put coordinate points comma r=4)
question
Find the distance between points B(5, 7) and F(1, 2), and the coordinates of the midpoint of BF.
answer
distance = ?41; 6.4 midpoint = (3, 9/2) Mathway
question
Find the standard equation for the ellipse, using the given characteristics. vertices: (0, ±8) foci: (0, ±?55)
answer
x²/9 + y²/64 = 1
question
Find: ?P?
answer
840
question
A hat contains 22 names, 10 of which are female. If five names are randomly drawn from the hat, what is the probability that at least four female names are drawn?
answer
0.105 ((??C? × ??C?)+(??C? × ??C?))÷(??C?) ((210 × 12)+(252 × 1))÷(26334) 2772÷26334 = 0.105
question
Teesha is in the bowling club. There are 33 students in the club. Five of them will be picked at random to attend an awards banquet. What is the probability that Teesha will not be randomly chosen to attend the banquet?
answer
28/33 Solution Probability of getting chosen is 5/33, so the the probability of not getting chosen is 33 - 5 = 28
question
How many different arrangements can be made using all of the letters in the word GAME?
answer
24 Solution ?P?
question
A spinner is evenly divided into 8 equal areas and numbered from 1 through 8. What is the probability of spinning a number less than 4 in a single spin?
answer
3/8
question
A lunch menu consists of 4 different kinds of sandwiches, 2 different kinds of soup, and 5 different drinks. How many choices are there for ordering a sandwich, a bowl of soup, and a drink?
answer
40 4 × 2 × 5 = 40
question
Two cards are randomly drawn in succession from a deck of 52 playing cards. Find the probability that the king of spades and any ace are drawn, in that order, without replacement.
answer
1/663 or 0.0015082 Solution 1/52 × 4/51 = 0.0015082
question
Evaluate: (??C? × ??C??)/??C?
answer
616 Solution ??C? = 792 ??C?? = 1001 ??C? = 1287 (792 × 1001)/1287 = 616
question
If 2 blocks are randomly taken from a bag containing 10 blue blocks, 10 red blocks, and 7 yellow blocks, what is the probability of drawing a blue block and a red block?
answer
50/351 or 0.1424501 Solution 10/27 × 10/26 = 0.1424501
question
Determine the probability that you will roll a number greater than 1 on a number cube.
answer
5/6
question
Evaluate: ?C?
answer
35
question
A circular, rotating, serving tray has 5 different desserts placed around its circumference. How many different ways can all of the desserts be arranged on the circular tray?
answer
24 Solution (n-1)! (5-1)! 4! = 4 × 3 × 2 × 1 = 24
question
How many different ways can 6 people be seated around a circular table?
answer
120 Solution (n-1)! (6-1)! 5! = 5 × 4 × 3 × 2 × 1 = 120
question
Three cards are drawn in succession and without replacement from a standard deck of 52 cards. How many sets of three cards are possible?
answer
22,100 Solution 52 choose 3 ??C? = 22,100
question
How many distinct committees of 14 people can be formed if the people are drawn from a pool of 23 people? Use factorials to express the answer.
answer
??C?? = 23!/(9! 14!) Solution nCr = n!/(r!(n-r)!) 23!/(14!(23-14)!) 23!/(14!(9)!) 23!/ (9! 14!)
question
Jamila spins a spinner with 3 sections of equal area, like the one below, 25 times. It lands on the 1eight times. What is the experimental probability of spinning a 1?
answer
8/25
question
In a bag there are 3 green jelly beans, 2 black jelly beans, and 8 yellow jelly beans. Once a jelly bean is drawn, it is not replaced. Find the probability of randomly drawing a green jelly bean and then a black jelly bean in two consecutive draws.
answer
1/26
question
Suppose you mix-up the cards below and choose one without looking. What is the probability of selecting neither "E" nor "G"? H E H E G W I
answer
4/7 Solution Probability of selecting "E" or "G" is 3/7. 7/7 - 3/7 = 4/7
question
Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 diamonds or 2 face cards?
answer
Incorrect: 0.108 Correct: 0.106 or 47/442 Solution (13/52 × 12/51) + (12/52 × 11/51) - (3/52 × 2/51)
question
A box contains 2 green, 6 yellow, and 3 purple balls. Find the probability of obtaining a purple ball in a single random draw.
answer
3/11
question
Find the four arithmetic means between -3 and 102.
answer
18, 39, 60, 81 Solution t(n) = t(1) + (n - 1)d 102 = -3 + (6 -1)d 102 = -3 + 5d 105 = 5d d = 21 Substitute t(2) = -3 + (2 - 1)21 -3 + (1)21 -3 + 21 18 t(3) = -3 + (3 - 1)21 -3 + (2)21 -3 + 42 39 And so on
question
Use Pascal's Triangle to determine the probability that you will get three green lights in a row of five lights. Assume red and green are equally likely occurrences.
answer
Incorrect: 7/32 Correct: 5/16 (10/32) Solution (n C k)(p?k)(1 - p)?(n-k) (?C?)(½)³(1 - ½)??³ (?C?)(½)³(1 - ½)² 10 × 1/8 × 1/4 = 5/16
question
Write the first five terms of the sequence defined by the given recursive or explicit formula. (Evernote)
answer
4, -3, 31, -167, 959
question
Expand the binomial raised to a power. (3a - b)?
answer
243a? - 405a?b + 270a³b² - 90a²b³ + 15ab? - b? Mathway: Expand Using the Binomial Theorem
question
Express 0.7312 as a geometric series, and write its sum as the ratio of two integers.
answer
(Evernote)
question
Evaluate the sum. (Evernote)
answer
-3003
question
Find the sum of the geometric series 0.2 + 0.02 + 0.002 + . . . given the formula S=a/(1 - r), where "a" is the first term, "r" is the common ratio, and "S" is the sum.
answer
2/9 Solution S = 0.2/(1 - 0.1) S = 0.222... or 2/9
question
Find the 6th and 8th entries in the row 12 of Pascal's triangle.
answer
792;792 Solution Use nCk-? to love for nth entry ??C?-? = ??C? = 792 ??C?-? = ??C? = 792
question
Use Pascal's triangle to find the number of ways to choose 3 boxes from 7 boxes.
answer
Incorrect: 21 Correct: 35 Solution ?C? = 35
question
Find the three positive geometric means between 6 and 1875/8.
answer
15, 75/2, 375/4 Solution t?=6 and t?=1875/8 t(n) = t?r^(n-1) t? = 6r? 1875/8 = 6r? r = 2.5 t? = 6(2.5)¹ 15 t? = 6(2.5)² 37.5 or 75/2 t? = 6(2.5)³ 9375 or 375/4
question
A 50-row theater has 30 seats in the front row. The second row has 31 seats. If each row has one more than the row in front of it, how many seats are there in the theater?
answer
2725 Solution t? = 30 and d = 1 t(n) = t? + (n-1)d t?? = 30 + (49)1 t?? = 79 S = n/2[2t? + (n - 1)d] S = 50/2[2(30) + 49] S = 25(60 + 49) S = 25(109) S = 2725
question
Find the sum of the infinite geometric series, if it exists. 3 - 9/5 + 27/25 - 81/125 + 243/625 - ...
answer
15/8 Solution (9/5) ÷ 3 = 0.6 r = 0.6 (common ration) t? = 3 S = t? ÷ (1 - r) S = 3 ÷ (1 - 0.6) S = 3 ÷ 0.4 S = 7.5 or 15/8
question
Use the given formula to find the first four terms of the arithmetic sequence. t(n) = 12 - 3n
answer
9, 6, 3, 0 Solution t(1) = 12 - 3(1) 9 t(2) = 12 - 3(2) 6 ...
question
There are 8 cars in a parking lot on a very cold day. Suppose the probability of any one of them not starting is 0.14. What is the probability that exactly 3 of the cars will not start?
answer
Incorrect: ? 0.003 Correct: ? 0.072 Solution (nCk) × (p)^n × (1 - p)^(n - k) (?C?) × (0.14)³ × (1 - 0.14)? = 0.072
question
Write an explicit formula for the nth term of the geometric sequence. 7/4, 49/24, 343/144, 2401/864, ...
answer
(Evernote) Solution explicit formula for nth term in a geometric sequence is t(n) = t?r^(n-1) t? = 7/4 and r = 7/6 t(n) = 7/4(7/6)^(n-1)
question
Find the sum of the infinite geometric series, if it exists. (Evernote) ?3(-½)^(k-1)
answer
2 Solution ?3(-½)^(k-1) t? = 3(-½)? 3 t? = 3(-½)¹ -1.5 r = t?/t? r = -1.5 ÷ 3 r = -½ S = t?/(1 - r) S = 3/(1 - (-½)) S = 3/1.5 S = 2
question
Find the sum of the first 7 terms of the geometric series 8 + 8/3 + 8/9 + 8/27 + ...
answer
11.99 Solution r ? 1 r = 8/3 ÷ 8 r = 1/3 & t? = 8 S(n) = t?[(1 - r^n) ÷ (1 - r)] S? = 8[(1 - (1/3)?) ÷ (1 - 1/3)] S? = 11.99
question
Use Pascal's triangle to solve for the value of n: 2(n C 3) = n+4 C ? (Evernote)
answer
Incorrect: 2 Correct: 4
question
Make a conjecture about the pattern of the given data. Find the sum of the 7th row. 3 3 + 6 + 3 3 + 6 + 9 + 6 + 3 3 + 6 + . . . + ( 3n ? 3) + ( 3n ) + ( 3n ? 3) + . . . + 3
answer
Incorrect: The sums of the rows are 3, 12, 27, . . . . The sum of the nth row is n(2n +3). The sum of the 7th row is 119. Correct: The sum of the rows are 3, 12, 27, . . . . The sum of the nth row is 3(n²). The sum of the 7th row is 147.
question
Find the two geometric means between 7 and 3584.
answer
Incorrect: 63, 567 Correct: 56, 448 Solution t?=7 and t?=3584 Formula t(n) = t?r^(n - 1) Substitute t(4) = 7r??¹ 3584 = 7r³ divide both sides by 7 r³ = 512 Take cube root of both sides r = 8 t(2)=7 × 8 56 t(3)=7 × 8² 448 7, 56, 448, 3584
question
Find the mean, the median, and mode of the data set: 14, 6, 21, 16, 16, 7, 20, 19, 7
answer
Mean: 14 Median: 16 Mode: 7, 16
question
Using the given information, which data set is the correct choice? (Evernote)
answer
20, 12, 27, 14, 16, 13, 12, 13, 26, 27, 21, 28, 20, 22, 20
question
Find the minimum and maximum values, quartiles, range, and interquartile range for the data set. 29, 14, 40, 12, 16, 46, 22, 23, 19, 31, 7, 9
answer
Minimum = 7 Maximum = 46 Q?=13; Q?=20.5; Q?=30 Range = 39 IQR = 17 Q? = mean
question
Last year, the personal best high jumps of track athletes in a nearby state were normally distributed with a mean of 205 cm and a standard deviation of 13 cm. What is the probability that a randomly selected high jumper has a personal best between 218 and 244 cm? (Evernote)
answer
0.1574 Solution (Graphing calculator) 2nd?VARS?DRAW?1:ShadeNorm(?Enter?75, 90, 85, 7) Between, mean, then standard deviation
question
The bowling scores of all of the bowlers in 12 bowling leagues are normally distributed. Their mean score is 201 points, with a standard deviation of 33 points. If one league has 120 bowlers, how many of them are likely to score more than 168 points?
answer
101 Solution (Graphing calculator) 2nd?VARS?DRAW?1:ShadeNorm(168, 10??, 201, 33)?Enter = 0.84 .84 or 84% 84% of 120 is 100.8 rounded to the nearest whole number equals 101
question
Find the variance and the standard deviation for the following data. 7, 20, 7, 15, 21, 4, 22, 17, 13
answer
variance ? 39.78; standard deviation ? 6.31 https://www.easycalculation.com/statistics/standard-deviation.php Take the Variance (Population Standard deviation) and Population Standard deviation
question
Find the minimum and maximum values, quartiles, range, and interquartile range for the data set. 27, 8, 9, 48, 19, 10, 29, 42, 2, 50, 44
answer
Minimum = 2 Maximum = 50 Q?=9; Q?=27; Q?=44 Range = 48 IQR = 35 Mathway Find the Five Number Summary, range, and Find the Interquartile Range (H-Spread)
question
Using the given stem and leaf plot and and it's median and mode, select the data set on which it was created. (Evernote)
answer
7, 6.9, 8.2, 7.6, 6, 7.2, 8.4, 7.6, 7.3, 7.8, 6.2, 9.5, 8.9, 6.5, 9
question
An English class was taken by 28 students. Their final scores were: three As, five Bs, nine Cs, six Ds, and five Fs. Using the given frequency table, find the mean grade point score, using A = 4, for the entire group of 28 students. Give the answer to the nearest hundredth, if necessary. (Evernote)
answer
Incorrect: [none] Correct: 1.82 (Close to a "C")
question
NCUse the frequency table below, with the a total of 18 catalog items sold at four different prices, to find the average price per item. (Evernote)
answer
$16.50
question
The relative frequency histogram below represents the age in years of the first 100 children to have their portraits taken at the "See What Develops" photography studio. What is the probability that the next child to have portraits taken will be between 1 and 2 years old?
answer
Incorrect: None of the listed answers are correct Correct: 31/100 or 31% (maybe 0.315 or 32%) Solution .11 + .20 + .31 + 30 + .15 + 24 = 1 (or 100%) Age 1= .20 Age 2= .11 .20 + .11 = .31 (or 31%) 31/100 or 31% Probability
question
A number cube is tossed 10 times with the following results. 5, 6, 1, 5, 5, 2, 6, 1, 2, 2 Find the range and the mean deviation of the data.
answer
None of the listed answers are correct To find mean deviation minus the mean from each data value, add everything up, then divide by the number of data values. Example: 54, 49, 47, 48, 52 mean of data = 50 54 - 50 = | 4 | = 4 49 - 50 = |-1| = 1 47 - 50 = |-3| = 3 48 - 50 = |-2| = 2 52 - 50 = | 2 | = 2 4 + 1 + 3 + 2 + 2 = 12 12 ÷ 5 = 2.4 Mean deviation = 2.4
question
A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of the purchases are returned. What is the probability that no more than 1 of the next 6 purchases will be returned?
answer
0.954 Solution ((?C?)(0.06)¹(1 - 0.06)?) + ((?C?)(0.06)?(1 - 0.06)?) = 0.954
question
In Sean's school there are 96 families which have 4 children. The circle graph shows the probability of each combination of girls and boys in a family with four children. Use the circle graph to predict the probability that one of these 96 families, chosen at random, will only have children of the same sex. (Evernote)
answer
12.5%
question
A biologist is collecting insects in a field. Beetles represent 67 percent of all the insects that have been collected so far. What is the probability that exactly half of the next 6 insects collected will be beetles?
answer
0.21617 Solution (?C?)(0.67)³(1-0.67)³ = 0.21617
question
The depth of snow at seven different mountain lodges is 13 in., 15 in., 21 in., 17 in., 90 in., 13 in., and 19 in. Find the mean, median, and mode. Tell which measure is the most useful for predicting how deep the snow will be at an eighth lodge.
answer
Mean = 26.9 in Median = 17 in Mode = 13 The median is the most useful.
question
The personal savings of the Young Saver Club were normally distributed with a mean of $525 and a standard deviation of $64. What is the probability that a randomly selected saver has an account total between $525 and $589?
answer
0.3413 Solution (Graphing calculator) 2nd?VARS?DRAW?1:ShadeNorm(525, 589, 525, 64)?Enter = 0.3413
question
A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of all purchases are returned. What is the probability that no more than 1 of the next 7 purchases will be returned?
answer
? 0.938 ((?C?)(0.06)¹(1 - 0.06)?) + ((?C?)(0.06)?(1 - 0.06)?) ? 0.938
question
The class average on a math test was 77.5 and the standard deviation was 5.8. Find the z-score for a test score of 83, and the percentage of the class who scored below 83.
answer
? 0.95; 82.89% Solution z-score z = (x - x?) ÷ ? z = (83 - 77.5) ÷ 5.8 = 0.948 ? 0.95 (Graphing calculator) 2nd?VARS?DRAW?1:ShadeNorm(-10??, 83, 77.5, 5.8)?Enter = 0.83 or 83%
question
In a certain normal distribution of scores, the mean is 30 and the standard deviation is 5.5. Find the z-score corresponding to a score of 37 and find the percentage of the scores that are below 37.
answer
1.27; 89.80% Solution z-score z = (x - x?) ÷ ? z = (37 - 30) ÷ 5.5 = 1.27 (Graphing calculator) 2nd?VARS?DRAW?1:ShadeNorm(-10??, 37, 30, 5.5)?Enter = 0.8984 or 89.80%
question
Refer to ? ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. Find tan x (Evernote)
answer
3/4 or 0.75 Solution tan x = opp./adj. 24/32 = 0.75
question
Evaluate the trigonometric expression. Sin?¹(-½)
answer
-30° Solution ? = 180° -?/6 -180°/6 = 30° Mathway Evaluate (Do not use Find the Exact Value)
question
Refer to ? ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. (Evernote)
answer
8/17 or 0.4706 sin? = opp./hyp.
question
Use the following information to find the unknown sides and angles. m?B = 28°; c=18
answer
b = 8.5; a = 15.9; m?A = 62°; m?C = 90° Solution We have two angles m?A + m?B + m?C = 180 m?A + 28° + 90° = 180° m?A = 62° Use Law of Sines to solve for side "a" or "b' a/sin A = c/sin C a/sin(62°) = 18/sin(90°) a = 18sin(62°)/sin(90°) a = 15.89 ? 15.9 b/sin B = a/sin A b/sin(28°) = 15.9/sin(62°) b = 15.9sin(28°)/sin(62°) b = 8.45 ? 8.5
question
What is the reference angle for 516° ?
answer
24° Mathway (Trigonometry) Find the Reference Angle
question
Find tan 421°.
answer
1.804 Mathway (Trigonometry) Find the Exact Value
question
For ? ABC, find the measure of ?A to the nearest degree. (Evernote)
answer
19° Solution Use Law of Cosines for "c" c = ?(a² + b² - 2ab cos C) c = 37 a = 12, b = 35, and c = 37 Use Law of Sines sin A/a = sin C/c sin A/12 = sin(90°)/37 sin A = 12sin(90°)/37 sin A ? 12/37 Convert from radians to degrees ?A = 18.58° ? 19°
question
Evaluate the trigonometric expression. cos(Cot?¹1)
answer
?2/2 Mathway (Trigonometry) Find the Exact Value
question
Given the quadrant of ? in standard position and a trigonometric function value of ?, find exact values for the remaining functions. Quadrant II, sin? = 2/7
answer
cos? = -3?5/7 tan? = -2?5/15 cot? = -3?5/2 sec? = -7?5/15 csc? = 7/2 Mathway (Trigonometry) Find the Other Trig Values in Quadrant II
question
Find tan 382°.
answer
0.404 Mathway (Trigonometry) Find the Exact Value
question
Convert 279° to radians.
answer
(31/20)? Mathway (Trigonometry) Convert from Degrees to Radians
question
Find the reference angle for 525°.
answer
15° Mathway (Trigonometry) Find the Reference Angle
question
Point P is located at the intersection of a circle with a radius of r and the terminal side of an angle ?. Find the coordinates of P to the nearest hundredth. ? = 120° , r = 13
answer
P(- 6.5, 11.26) Solution 120° is in Quadrant II Coordinates: (cos ? = x/r, sin ? = y/r) ? = 120° and r = 13 cos(120°) = x/13 -½ = x/13 x = -13/2 or -6.5 sin(120°) = y/13 ?3/2 = y/13 y = 13?3/2 or 11.26 P(-6.5, 11.26)
question
The function d = - 6 cos 3t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the frequency?
answer
Incorrect: 3?/2 cycles/seconds Correct: 3/2? Solution d = A cos wt d = -6 cos 3t w = 3 f (frequency) = w/2? f = 3/2?
question
For a circle of radius 4 feet, find the arc length "s" subtended by a central angle of ?/30 radians. Round to the nearest hundredth.
answer
None of the listed answers are correct Solution r = 4 and ? = ?/30 s = r? s = 4(?/30) s = 2?/15
question
The function d = 9 cos 2t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the maximum displacement of the object from its resting position?
answer
9 m Solution A = displacement d = A cos wt d = 9 cos 2t A = 9 or 9m
question
Find cot (-290°).
answer
0.364 Mathway (Trigonometry) Find the Exact Value
question
For a circle of radius 8 feet, find the arc length of a central angle of 30°.
answer
(4/3)? feet Solution r = 8 and ? = ?/6 (change 30° to radians) s = r? s = 8(?/6) s = (4/3)? or 4?/3
question
Find the amplitude, the period, and the frequency of the graph. Then write an equation for the sine function for the graph.
answer
3/4, 180°, 2, y = 3/4 sin 2x Graph all multiple choices and pick the one that fits
question
Find the exact value of the sine, cosine, and tangent of -210°.
answer
sin = 1/2 cos = -?3/2 tan = -?3/3 Mathway (Trigonometry) Find the Exact Value
question
Suppose the depth of the tide in a certain harbor can be modeled by y = 25 - 4 cos(?/6t), where "y" is the water depth in feet and "t" is the time in hours. Consider a day in which t = 0 represents 12:00 midnight. For that day, when are high tide and low tide and what is the depth of each?
answer
Incorrect: high tide at 12:00 noon and 12:00 midnight, depth 29 ft low tide at 6:00 a.m. and 6:00 p.m., depth 21 ft Correct: high tide at 12:00 noon and 12:00 midnight, depth 21 ft low tide at 6:00 a.m. and 6:00 p.m., depth 29 ft
question
Use either the Law of Sines or the Law of Cosines to solve for "a" to the nearest tenth. (Evernote)
answer
28.8 Solution ?A + 61° + 76° = 180° ?A = 43° Use Law of Sines a/sin(43°) = 41/sin(76°) a = 41sin(43°)/sin(76°) a = 28.8
question
Find cos((1/2)A) and sin((1/2)A) if sin(A)=4/5 and Ais a first-quadrant angle. (Evernote)
answer
Incorrect: cos(A/2) = ?3/5 sin(A/2) = 2/5 Correct: cos(A/2) = (2?5)/5 sin(A/2) = ?5/5 Solution Pathagorean Identities sin A = 4/5 cos²A = 1 - sin²A cos A = 1 - (4/5)² cos A = 3/5 Use Half-Angle Identities sin(A/2) = ?((1 - cos A)/2) sin(A/2) = ?((1 - 3/5)/2) sin(A/2) = ?5/5 cos(A/2) = ?((1 + cos A)/2) cos(A/2) = ?((1 + 3/5)/2) cos(A/2) = (2?5)/5 A = ? and the cos((1/2)A) is the same as cos(A/2)
question
Find all solutions of cos(x) - ?(1 - 3cos²(x)) = 0
answer
Incorrect: None of the listed answers are correct Correct: No Solution
question
Use the Law of Sines to solve for m?A to the nearest tenth. (Evernote)
answer
29.8 Solution sin B/b = sin C/c sin B/43 = sin(80°)/45 sin B = 43sin(80°)/45 sin B = 0.94103 sin?¹(sin B) = ?B sin?¹((43sin¹(80°)/45)) sin?¹(0.94103) = 70.2° (rounded to the nearest tenth) ?B = 70.2° ?A + 70.2° + 80° = 180° ?A = 29.8° Use graphing calculator on degree mode or use Mathway and change answer to degrees.
question
Write the expression csc(2x) + cot(2x) in terms of tan x or cot x.
answer
Incorrect: None of the listed answers are correct Correct: cot(x)
question
Use the Law of Cosines to solve for "A" to the nearest tenth of a degree. (Evernote)
answer
35.3° Law of Cosines cos A = (b² + c² - a²)/(2bc) cos A = (43² + 50² - 29²)/(2 × 43 × 50) cos A = 0.8158 cos?¹(0.8158) = 35.3° Just do this ? ?A = cos?¹((b² + c² - a²)/(2 × 43 × 50)) ?A = 35.3°
question
Find the exact value of sin(- 450° + 300°).
answer
-1/2 Solution Use Sum and Diffrence Identities: sin(A + B) = sinAcosB + cosAsinB sin(450°)cos(300°) + cos(450°)sin(300°) = -1/2 Mathway (Trigonometry) Find the Exact Value
question
Solve sin(x) - ?(1 - 3sin²(x)) = 0 given that 0° ? x ? 360°.
answer
Incorrect: 90°, 270° Incorrect: None of the listed answers are correct Correct: 30°, 150°, 210°, 330° Solution sin(x) = ?(1 - 3sin²(x)) sin²(x) = 1 - 3sin²(x) sin²(x) + 3sin²(x) - 1 4sin²(x) - 1 (2sin(x) + 1)(2sin(x) - 1) sin(x) = -1/2 and 1/2 x = -30°, 30°, 150°, 210° Add 360° to -30° -30° + 360° = 330° x = 30°, 150°, 210°, 330° Or graph y = sin²(x) y = 1 - 3sin²(x) Intersections: 30°, 150°, 210°, 330°
question
Find the image of (5, -6) after a counterclockwise rotation of 270° about the origin. (Evernote)
answer
(-6, -5)
question
Wilfred is checking his bicycle's wheel. The wheel has a radius of 25 cm, and its center is 54 cm above the ground. Wilfred is rotating the wheel at a constant speed of 230° /s. The height of a point on the tire as a function of time is given by h = 54 + 25 sin(230t), where h is the height in centimeters and t is the time in seconds. Find h when t = 5.5 s. Round your answer to the nearest tenth of a centimeter.
answer
Incorrect: None of the listed answers are correct Correct: 51.8 Solution h = 54 + 25 sin(230t) h = 54 + 25 sin(230 × 5.5) h = 51.8
question
Solve sin(x) - ?(1 - 3sin²(x)) = 0 given that 0° ? x ? 360°
answer
Incorrect: 90°, 270° Incorrect: None of the listed answers are correct Correct: 30°, 150°, 210°, 330° Solution sin(x) = ?(1 - 3sin²(x)) sin²(x) = 1 - 3sin²(x) sin²(x) + 3sin²(x) - 1 4sin²(x) - 1 (2sin(x) + 1)(2sin(x) - 1) sin(x) = -1/2 and 1/2 x = -30°, 30°, 150°, 210° Add 360° to -30° -30° + 360° = 330° x = 30°, 150°, 210°, 330° Or graph y = sin²(x) y = 1 - 3sin²(x) Intersections: 30°, 150°, 210°, 330°
question
Given that F = 32°, G = 55°, and f = 10, solve the triangle. If no such triangles exist, write none. Round to the nearest tenth. (Evernote)
answer
E = 93° , g = 15.5, e = 18.8 Solution F = 32°, G = 55° , E = ?° f = 10, g = ?, e = ? 32° + 55° + ?E = 180° ?E = 93° Use Law of Sines 10/sin(32) = g/sin(55) g = 10sin(55)/sin(32) g = 15.5 10/sin(32) = e/sin(93) e = 10sin(93)/sin(32) e = 18.8
question
Given sin? = 5/9, where ?/2 < ? < ?, find the exact values of sin 2? and cos 2?.
answer
Incorrect: -31/81, (20?14)/81 Correct: 10/9, -(4?14)/9
question
Which of the following is equal to sin²(x)/(1 - cos(x))
answer
Incorrect: None of the listed answers are correct Correct: 1 + cos(x) or 1 + cos? Mathway Rationalize the Denominator Mathway Verify the Identity+
question
Find the exact value of cos(345°).
answer
(?6 + ?2)/4 Mathway (Trigonometry) Find the Exact Value
question
Whitney starts the engine on her small private airplane. The engine drives a propeller with a radius of 7.5 feet and its centerline 12.5 feet above the ground. At idle, the propeller rotates at a constant speed of approximately 850 revolutions per minute. The height of one propeller tip as a function of time is given by h = 12.5 + 7.5 sin(850t), where h is the height in feet and t is the time in minutes. Find h when t = 2.5
answer
8.2 ft Solution h = 12.5 + 7.5 sin(850t) h = 12.5 + 7.5 sin(850 × 2.5) h = 8.2
question
Solve triangle ABC given that a = 11, b = 13, and c = 15.
answer
A = 45.6° , B = 57.6° , C = 76.9° Solution Use Law of Cosines cos A = (b² + c² - a²)/2bc cos A = (13² + 15² - 11²)/(2 × 13 ×15) cos A = 7/10 cos?¹(7/10) = 45.6° cos B = (c² + a² - b²)/2ca cos B = (15² + 11² - 13²)/(2 × 15 × 11) cos B = 59/110 cos?¹(59/110) = 57.6° cos C = (a² + b² - c²)/2ab cos C = (11² + 13² - 15²)/(2 × 11 × 13) cos C = 5/22 cos?¹(5/22) = 76.9°
question
csc(2x) + cot(2x) forms an identity with which of the following?
answer
cot x Use Mathway to "Verify the Identity" csc(2x) + cot(2x) = cot(x)
question
Which of the following is an identity?
answer
(cos(x))/(1 + sin(x)) = (1 - sin(x))/(cos(x)) Mathway Verify the Identity
Get an explanation on any task
Get unstuck with the help of our AI assistant in seconds
New