[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above are satisfied.}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

a) The maximum point touches the $y$-axis. Thus, we shall use cosine function.

period: $T=90^circimplies k = dfrac{360}{90}=4$

amplitude: $A=dfrac{|8-4|}{2}=2$

equation of axis: $f(x)=dfrac{8+4}{2}=6$

Therefore, the equation is

$$
f(x)=2cos (4x)+6
$$

b) The minimum point touches the $y$-axis. Thus, we shall use cosine function but we need to reflect it over $x-$axis .

period: $T=180^circimplies k = dfrac{360}{180}=2$

amplitude: $A=dfrac{|3-1|}{2}=1$

equation of axis: $f(x)=dfrac{3+1}{2}=2$

Therefore, the equation is

$$
f(x)=-cos(2x)+2
$$

b) The maximum point touches the $y$-axis. Thus, we shall use cosine function.
period: $T=120^circimplies k = dfrac{360}{120}=3$

amplitude: $A=dfrac{|0-(-4)|}{2}=2$

equation of axis: $f(x)=c=dfrac{0+(-4)}{2}=-2$

Therefore, the equation is

$$
f(x)=2cos(3x)-2
$$

a) $f(x)=2cos(4x)+6$

b) $f(x)=-cos(2x)+2$

c) $f(x)=2cos(3x)-2$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

From the table

Maximum occurs at $x=0$ ($y-$axis) $implies$ cosine function with $A>0$

period: $T=180^circimplies k=dfrac{360}{180}=2$

$y_{text{min}}=5$

$y_{text{max}}=9$

amplitude: $A=dfrac{|9-5|}{2}=2$

axis: $c=dfrac{9+5}{2}=7$

Using these information, the equation that describes the data is

$$
y=2cos(2x)+7
$$

$$
y=2cos(2x)+7
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

In this case,

$$
A=4implies c=9-4=5
$$

$T=120^circ implies k= dfrac{360^circ}{120^circ}=3$

maximum at $(0,9)implies$ cosine function with $A>0$

Thus, the equation is

$$
y=4cos(3x)+5
$$

$$
y=4cos(3x)+5
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

a)

(i) None of the conditions mentioned above applies, but notice that the minimum point would have touched the $y-$axis if it wasn’t translated 1 unit to the right. So here, we shall use cosine function with $A<0$ and $d=1$ (horizontally translated 1 unit to the right)

period = $6implies k=dfrac{360}{6}=60$

amplitude = $A=dfrac{8-2}{2}=3$

axis: $c=dfrac{8+2}{2}=5$

Therefore

$y=-Acos[k(x-d)]+c$

$$
y=-3cos[60(x-1)^circ]+5
$$

(ii) The minimum touches the $y-$axis, thus, we shall use cosine function with $A<0$

period = 3 $implies k=dfrac{360}{3}=120$

amplitude: $A=dfrac{1.5-0.5}{2}=0.5$

axis: $c=dfrac{1.5+0.5}{2}=1$

Therefore

$$
y=-0.5cos(120x)^circ+1
$$

(iii) The axis (half-way between maximum and minimum) touches the $y-$axis and is decreasing as it cross the $y-$axis. Thus, we shall use sine function with $A<0$

period = $4implies k =dfrac{360}{4}=90$

amplitude: $A=dfrac{|-3-(-1)|}{2}=1$

axis: $c=dfrac{(-3)+(-1)}{2}=-1$

Therefore

$$
y=-sin(90x)^circ-1
$$

b)

(i) The axis (half-way between maximum and minimum) touches the $y-$axis and is decreasing as it cross the $y-$axis. Thus, we shall use sine function with $A<0$

period = $4implies k =dfrac{360}{4}=90$

amplitude: $A=dfrac{|30-20|}{2}=5$

axis: $c=dfrac{(30)+(20)}{2}=25$

Therefore

$$
y=-sin(90x)^circ-1
$$

(ii) The maximum point would have touched the $y$-axis if it was not translated 1 unit to the left. Thus, we shall use cosine function with $A>0$ and $d=-1$ (horizontal shift of 1 unit to the left).

period: $T=3implies k=dfrac{360}{3}=120$

amplitude: $A=dfrac{|15-5|}{2}=5$

axis: $c=dfrac{15+5}{2}=10$

Therefore,

$$
y=5cos[120(x+1)^circ]+10
$$

(iii) The maximum point touches the $y-$axis. Thus, we shall use cosine function with $A>0$

period: $T=1implies k =360$

amplitude: $A=dfrac{5-(-15)}{2}=10$

axis: $c=dfrac{5-15}{2}=-5$

Therefore,

$$
y=10cos(360x)^circ-5
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

a)

Maximum point occurs at the $y$-axis (at $x=0)$, thus, we shall use cosine function with $A>0$.

period = $120^circimplies k=dfrac{360}{120}=3$

amplitude = $A=dfrac{3-1}{2}=1$

axis: $c=dfrac{3+1}{2}=2$

Therefore

$$
y=cos(3x)^circ+2
$$

b)

period = $720^circ implies k =dfrac{360}{720}=0.5$

amplitude = $A=dfrac{21-13}{2}=4$

axis: $c=dfrac{21+13}{2}=17$

The point halfway between maximum and minimum would have touched the $y$-axis in an increasing manner if it wasn’t translated 180$^circ$ to the right. Thus, we shall use sine function with $A>0$ and $d=180^circ$.

$y=4sin[0.5(x-180)^circ]+17$

Since the maximum point is located $360^circ$ to the right of $y$-axis, we can also use

$$
y=4cos[0.5(x-360)^circ]+17
$$

c)

period = $180-(-60)=240implies k=dfrac{360}{240}=1.5$

amplitude= $A=dfrac{|-1-(-7)|}{2}=3$

axis: $c=dfrac{(-7)+(-1)}{2}=-4$

The point halfway between maximum and minimum crosses the $y-$intercept and is increasing as it crosses, thus we shall use sine function with $A>0$.

$$
y=3sin(1.5x)^circ-4
$$

d)

period $=130-10=120implies k=dfrac{360}{120}=3$

amplitude = $A=dfrac{|5-(-1)|}{2}=3$

axis: $c=dfrac{5+(-1)}{2}=2$

The maximum point $(5,10^circ)$ is translated $10^circ$ to the right from the $y$-axis. Thus, we shall use cosine function with $A>0$ and $d=10$

$$
y=3cos[3(x-10)]+2
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

Here we need to use only cosine function only. Parameters $A, d, c$ are already given, but we have to calculate $k$ from the given period $T$. Remember that

$k=dfrac{360^circ}{T}$

a) $k=dfrac{360}{360}=1implies y=3cos x+11$

b) $k=dfrac{360}{180}=2implies y=4cos[2(x-30)^circ]+15$

c) $k=dfrac{360}{40}=9implies y= 2cos[9(x-7)^circ]$

d) $k=dfrac{360}{720}=0.5implies y= 0.5cos[0.5(x+56)^circ]-3$

a) $y=3cos x +11$

b) $y=4cos[2(x-30)^circ]+15$

c) $y=2cos[9(x-7)^circ]$

d) $y=0.5cos[0.5(x+56)^circ]-3$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

In this case, $A=6$, $T=45^circ implies k =dfrac{360}{45}=8$, $c=1+6=7$

The minimum value is $(0,1)$ so the minimum point touches the $y$-axis. Thus, we shall use cosine function with $A<0$

Therefore

$$
y=-6cos(8x)^circ+7
$$

$$
y=-6cos(8x)+7
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

b) The graph is periodic and sinusoidal, so it can be modeled by a trigonometric function. The minimum occurs at $x=0$, we can model this by a cosine function.

c) The period is $12$ months, $k=dfrac{360}{12}=30$

amplitude = $A=dfrac{17-(-18.6)}{2}=17.8$

axis: $c=dfrac{17+(-18.6)}{2}=-0.8$

The minimum point touches the $y$-intercept, so it is a cosine function with $A<0$

$$
y=-17.8cos(30x)-0.8
$$

d) We shall find $y$ when $x=20$

$y=-17.8cos[30(20)]-0.8$

$$
y=8.1
$$

a) see graph

b) trigonometric function, cosine function

c) $y=-17cos(30x)-0.8$

d) $y=8.1$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

b)

amplitude: $A=dfrac{0.85-0}{2}=0.425$

axis: $c=dfrac{0.85+0}{2}=0.425$

period: $T=3implies k =dfrac{360}{3}=120$

The minimum point is the $y$-intercept, so we can use cosine function with $A<0$

$y=-0.425cos(120t)+0.425$

c)

d) Since the period is $3$s, we expect that at $6$s, the velocity is $0$ L/s.

e) The time that corresponds to $v=0.5$ L/s is

a) breathing occurs at regular intervals

b) $y=-0.425cos(120t)^circ+0.425$

c) see graph

d) $0$ L/s

e) $t=0.835s,; 2.165s,; 3.835s$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

All three graphs has a minimum as the $y$-intercept, thus we will use a cosine function with $A<0$. All also have a period of 12 months (1 year) $implies k =dfrac{360}{12}=30$.
b)

Athens: $y_{min}=12$ , $y_{max}=33$

amplitude = $A=dfrac{|33-12|}{2}=10.5$

axis: $c=dfrac{33+12}{2}=22.5$

$y=-10.5cos(30t)^circ+22.5$

Lisbon: $y_{min}=13$ , $y_{max}=27$

amplitude: $A=dfrac{|27-13|}{2}=7$

axis: $c=dfrac{27+13}{2}=30$

$y=-7cos(30t)^circ+30$

Moscow: $y_{min}=-9$ , $y_{max}=23$

amplitude: $A=dfrac{|23-(-9)|}{2}=16$

axis: $y=dfrac{23+(-9)}{2}=7$

$$
y=-16cos(30t)^circ+7
$$

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

a) The point halfway between maximum and minimum point crosses the $y$-axis in increasing manner, thus, we shall use sine function with $A>0$

From the graph

amplitude = $A=11-8=3$

axis: $c=8$

period = $0.04implies k =dfrac{360^circ}{0.04}=9000$

Therefore, the equation that describes the graph is

$$
y=3sin(9000t)^circ+8
$$

b) The peak represents the maximum stress in MPa on the shaft of an electric motor.

c) We shall find the value of $y$ when $t=0.143$

$y=3sin(9000cdot 0.143)^circ+8approx 6.64$ MPa

a) $y=3sin(9000t)^circ+8$

b) maximum stress on the shaft

c) 6.64 MPa

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{cosine function: the maximum or minimum point is the $y-$intercept}} hfill \
{text{If the maximum point touches the $y$-intercept, A $>$ 0}} hfill \
{text{If the minimum point touches the $y$-intercept, A$$ 0}}hfill \
{text{If the graph is decreasing as it crosses $y-$axis, A $<$ 0}}hfill \\
{text{If none of the above conditions is satisfied}} hfill \
{text{perform necessary horizontal translations}} hfill \
{text{so that one of the conditions above is satisfied.}}hfill \\
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{If the minimum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=-Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

Initially (at $x$=0) where the tire just picked up the nail, the nail is at the ground, which is the minimum point. The minimum point is not translated from the $y$-axis, thus, we shall use $y=-Acos(kx)+c$

The amplitude correspond to the radius of the tire, which is $r=dfrac{60text{}}{2}=30$ cm

Since the ground is at $y=0$, the axis is the height of the center of the wheel from the ground which is also the radius, thus, $c=30$

The period is the distance traveled in one revolution is equal to the circumference of the wheel which is

$T=60pi implies k=dfrac{360^circ}{60pi}approx1.90986$.

Therefore, the equation describing the height of the nail as a function of time is

$y=-30cos(1.90986 x)^circ+30$

Thus, at $x=1;text{km}times dfrac{1000;text{m}}{1;text{km}}times dfrac{100;text{cm}}{1;text{m}}=1times 10^5$ cm

$y=-30cos(1.90986times 10^5)+30=59.84$ cm

Therefore, when the car traveled 1 km, the nail is 59.84 cm from the ground.

$59.84$ cm

[begin{gathered}
{text{For a sinusoidal function, remember the following }} hfill \
hfill \
y = Acos left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
{text{or }} hfill \
y = Asin left[ {kleft( {x – d} right)} right] + c{text{ }} hfill \
hfill \
{text{amplitude = }}|A| = dfrac{|y_{text{max}}-y_{text{min}}|}{2}hfill \
{text{period = }}T=frac{{{{360}^ circ }}}{|k|} hfill \
{text{equation of axis:}}{text{ }},,,y = c = dfrac{y_{text{max}}+y_{text{min}}}{2} hfill \
hfill \
{text{Generally, if the maximum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=Acos[k(x-d)^circ]+c$}}hfill \\
{text{If the minimum point is translated $d$ units to the right of the}}hfill \\
{text{$y$-axis, the equation is $y=-Acos[k(x-d)^circ]+c$}}hfill \\
{text{Note that multiple possible sinusoidal functions can describe a graph or data}}hfill \
end{gathered} ]

Initially (at $t$=0) Matthew is at the highest point of the Ferris wheel. The maximum is not translated from the $y$-axis, thus, we shall use $y=Acos(kx)+c$

The amplitude correspond to the radius of the tire, which is $r=dfrac{60text{}}{2}=7$ m

Since the boarding height of the wheel is 1m, the axis is the radius of the wheel plus 1 m, which means $c=8$.

The period is the time it takes for one revolution which is equal to the circumference of the wheel divided by the speed (10 km/h) which we shall convert to m/s.

$T=dfrac{2pi times 7;text{m}}{10,000;text{m}/h times dfrac{1;text{h}}{3600;text{s}}}=15.83$ s

This means $k=dfrac{360}{15.83}=22.74$

Therefore

$y=7cos(22.74t)^circ+8$ where $y$ is in meters and $t$ is in seconds

$$
y=7cos(22.74t)^circ+8
$$