HW #32: Related Rates Practice
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1. A conical paper cup 3 inches across the top and 4 inches deep is full of water. The cup springs a leak at the bottom and loses water at the rate of 2 cubic inches per minute. How fast is the water level dropping at the instant when water is exactly 3 inches deep? Express the answer in inches per minute.
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(Hint: Don't forget to write radius in terms of height before differentiating!) dh/dt = -128/(81pi) in/min
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2. Air is being pumped into a spherical balloon at the rate of 7 cubic centimeters per second. What is the rate of change of the radius at the instant the volume equals 36π?
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dr/dt=7/(36pi) cm/sec
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3. A kite 100 feet above the ground is being blown away from the person holding its string in a direction parallel to the ground at the rate of 10 feet per second. At what rate must the string be let out when the length of the string already let out is 200 feet?
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(Hint: Think about what is constant here. You can plug in that value before differentiating) ds/dt = 5(square root 3) ft/sec
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4. All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is 10 centimeters?
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(Hint: V=x^3 is the equation used because it is a cube) dV/dt = 900 cm^3/sec
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5. A kite is flying at an angle of elevation of π/3. The kite string is being taken in at the rate of 1 foot per second. If the angle of elevation does not change, how fast is the kite losing altitude?
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(Hint: Use sine to relate the distance (hypotenuse) and height (leg) of the kite.) dh/dt = (square root of 3)/2 ft/sec
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6. Sand is dumped off a conveyor belt into a pile at the rate of 2 cubic feet per minute. The sand pile is shaped like a cone whose height and base diameter are always equal. At what rate is the height of the pile growing when the pile is 5 feet high?
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(Hint: Volume equation should be written only in terms of h before differentiating. How can you relate radius and height?) dh/dt = 8/25pi ft/min
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7. A camera is located 50 feet from a straight road along which a car is traveling at 100 feet per second. The camera turns so that it is pointed at the car at all times. In radians per second, how fast is the camera turning as the car passes closest to the camera?
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(Hint: Theta is not given, but you know what it will be. When the car passes by, the \"hypotenuse\" will overlap with a \"leg,\" meaning the angle is...? Also, use tangent to link all variables.) d(theta)/dt = -2 rad/sec
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8. A balloon leaves the ground 500 feet away from an observer and rises vertically at the rate of 140 feet per minute. At what rate is the angle of inclination of the observer's line of sight increasing at the instant when the balloon is exactly 500 feet above the ground?
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(see guided notes) D(theta)/dt = 7/50 rad/sec
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9. A spherical balloon with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 30 centimeters?
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dr/dt = 2/(9pi) cm/sec
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10. A ladder 13 feet long is leaning against the side of a building. If the foot of the ladder is pulled away from the building at a constant rate of 2 inches per second, how fast is the angle formed by the ladder and ground changing (in radians per second) at the instant when the top of the ladder is 12 feet above the ground?
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d(theta)/dt = -1/72 rad/sec
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11. A ladder 13 feet long is leaning against the side of a building. If the foot of the ladder is pulled away from the building at a constant rate of 8 inches per second, how fast is the area of the triangle formed by the ladder, the building, and the ground changing (in feet squared per second) at the instant when the top of the ladder is 12 feet above the ground?
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(Hint: When you differentiate area, it becomes obvious that you will need to dy/dt in order to get dA/dt. First use x^2 +y^2 = 13^2 to find dy/dt and then plug that in to dA/dt = .5xdy/dt +.5ydy/dt.) dy/dt = -5/18 ft/sec dA/dt = 119/36 ft^2/sec
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12. Truck A travels east at 40 mph and Truck B travels north at 30 mph. How fast is the distance between the trucks changing 6 minutes later?
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dC/dt = 50 mph
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13. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
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dA/dt = 8pi ft^2/sec
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14. A student release helium from a spherical balloon at a rate 2 centimeters per second. How fast is the radius of the balloon decreasing at the instant when the radius is 18 centimeters?
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dr/dt = -1/(18^2pi) cm/sec
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15. An air traffic controller spots 2 planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 150 miles away from the point moving at 450 miles per hour. The other plane is 200 miles from the point moving at 600 miles per hour. At what rate is the distance between the planes decreasing? How much time does the air traffic controller have to get one of the planes on a different path?
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dC/dt = -750 mph time = 20 minutes
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16. At a sand and gravel plant, sand is falling off a conveyer belt and onto a conical pile at a rate of 10 cubic feet per minute. At what rate is the height of the pile changing when the pile is 15 feet high and the diameter of the base is 40 ft?
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dh/dt = 3/8pi ft/min
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17. A vertically rising rocket is tracked by radar 5 miles away from the launch site. How fast is the rocket rising when it is 4 miles high and its distance from the radar station is increasing at a rate of 2000 mph?
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dy/dt = 500(square root of 41) mph
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18. Grain falling through a chute at 10 cubic feet per min creates a conical pile whose height always equals its diameter. How fast will the circumference of the base changing when the pile's height is 8 feet?
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(Hint: First you need to find dr/dt using the volume formula. Then you will use the circumference formula.) dr/dt = 5/(16pi) ft/min dC/dt = 5/8 ft/min
