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Related Rate Exercises I 1) A ladder 26ft long rests on a horizontal ground and leans against a vertical wall. The foot of the ladder slides out at a rate of 4ft/s. How fast is the top of the ladder sliding down the wall when the foot is 10ft. from the wall? 2) A person on a pier pulls in a boat using a rope fastened to the bow. The person pulls in on the rope at 5ft/min and is holding the rope 15ft above the fastening. How fast is the boat moving when it is 20ft away from the peir? 3) Car A is traveling west at 50km/h, car B is traveling north at 60km/h and both cars are headed toward the intersection of two roads. At what rate are the cars approaching each other when car A is 0.3km and car B is 0.4km from the intersection? 4) At a certain instant the length of a rectangle is 16cm long, the width is 12cm and the width is increasing at a rate of 3cm/s. How fast is the length changing if the area of the rectangle is increasing at the rate of 24cm 2 /s? 5) A cylindrical tank is 6m high, has a radius of 2m and is being filled at a rate of 1/100m 3 /min. How fast is the water rising when the water is 0.5m deep? 6) A spherical balloon is being inflated with air and the volume of the air in the balloon is increasing at 20cm 3 /sec. When the radius is 30cm, how fast is the radius of the balloon increasing at that time? 7) A person 6ft tall walks at the rate of 5ft/s toward a street light that is 16ft above the ground. At what rate is the length of the shadow changing when the person is 10ft from the base of the light? At what rate is the tip of the person’s shadow moving? 8) A person is 5ft tall and walks away from a 20ft tall lamp post at 7ft/sec. When the person is 8ft from the lamp post calculate the rate at which the tip of the person’s shadow is moving. 9) A water tank is in the shape of a cone having a base radius of 2m and a height of 4m. If water is pumped in at a rate of 2m 3 /min, find the rate at which the water level is rising when the water is 3m deep. 10) Gravel is being dumped from a conveyor belt at a rate of 30ft 3 /min. The gravel forms a conical pile with base diameter and height being equal. How fast is the pile increasing when the pile is 10ft high? 11) A balloon is released at a point A and rises vertically with a constant speed of 5m/s. Point B is on the ground at a distance of 10m from point A. How fast is the angle of elevation of the balloon at point B changing when the balloon is 20m above the ground? 12) Water runs into a conical tank at a rate of 20ft 3 /min. The tank has a height of 10ft and a base radius of 15ft. How fast is the water rising when the water is 6ft deep?

Related Rates All Qs and Answers

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Page 1: Related Rates All Qs and Answers

Related Rate Exercises I 1) A ladder 26ft long rests on a horizontal ground and leans against a vertical wall.

The foot of the ladder slides out at a rate of 4ft/s. How fast is the top of the ladder sliding down the wall when the foot is 10ft. from the wall?

2) A person on a pier pulls in a boat using a rope fastened to the bow. The person

pulls in on the rope at 5ft/min and is holding the rope 15ft above the fastening. How fast is the boat moving when it is 20ft away from the peir?

3) Car A is traveling west at 50km/h, car B is traveling north at 60km/h and both

cars are headed toward the intersection of two roads. At what rate are the cars approaching each other when car A is 0.3km and car B is 0.4km from the intersection?

4) At a certain instant the length of a rectangle is 16cm long, the width is 12cm and

the width is increasing at a rate of 3cm/s. How fast is the length changing if the area of the rectangle is increasing at the rate of 24cm2/s?

5) A cylindrical tank is 6m high, has a radius of 2m and is being filled at a rate of

1/100m3/min. How fast is the water rising when the water is 0.5m deep? 6) A spherical balloon is being inflated with air and the volume of the air in the balloon

is increasing at 20cm3/sec. When the radius is 30cm, how fast is the radius of the balloon increasing at that time?

7) A person 6ft tall walks at the rate of 5ft/s toward a street light that is 16ft above

the ground. At what rate is the length of the shadow changing when the person is 10ft from the base of the light? At what rate is the tip of the person’s shadow moving?

8) A person is 5ft tall and walks away from a 20ft tall lamp post at 7ft/sec. When the

person is 8ft from the lamp post calculate the rate at which the tip of the person’s shadow is moving.

9) A water tank is in the shape of a cone having a base radius of 2m and a height of

4m. If water is pumped in at a rate of 2m3/min, find the rate at which the water level is rising when the water is 3m deep.

10) Gravel is being dumped from a conveyor belt at a rate of 30ft3/min. The gravel

forms a conical pile with base diameter and height being equal. How fast is the pile increasing when the pile is 10ft high?

11) A balloon is released at a point A and rises vertically with a constant speed of

5m/s. Point B is on the ground at a distance of 10m from point A. How fast is the angle of elevation of the balloon at point B changing when the balloon is 20m above the ground?

12) Water runs into a conical tank at a rate of 20ft3/min. The tank has a height of 10ft

and a base radius of 15ft. How fast is the water rising when the water is 6ft deep?

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Solutions-Related Rate Exercise I
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Solutions Related Rate Exercise I
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Related Rate Exercise II
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Related Rates

In many real world problems two or more related quantities are changing with time t. Related rate problems involve calculating the rate at which some quantity is changing with respect to time given the rates of change of some other quantities. An equation relating these quantities is first determined and then is implicitly differentiated with respect to time producing an equation relating the derivatives of the quantities. The general approach for solving related rate problems is as follows:

• Draw a diagram and assign variables to the quantities • Determine an equation relating these quantities • Implicitly differentiate the equation with respect to time • Substitute the given values into the equation and solve for the desired rate of

change at the given instant of time 1) Gas is pumped into a spherical balloon at the rate of 50ft3/s. How fast is the radius of

the balloon increasing at the instant when the radius is 12ft? How fast is the surface area increasing at the same instant? {(25/288π)ft/sec, 25/3ft2/s}

2) Water floats out of a conical container at a rate of 3in3/s. The container has a radius of

2in and a height of 8in. How fast is the water level dropping at the instant when the level is 5in high? {(-48/25π)in/sec}

3) A 20ft ladder leans against a vertical wall. If the base of the ladder is pulled away

from the wall at the constant rate of 4ft/s, find the speed at which the top of the ladder is sliding down the wall at the instant when the base is 12ft from the wall. {-3ft/s}

4) A person 1.8m tall is walking away from a lamp post that is 4m high at the rate of

2m/s. At what rate is the person’s shadow lengthening when the person is 6m away from the lamp post? {18/11m/s}

5) A car leaves point A and travels north at a speed of 20mph. Two hours later a second

car heads east from A at 30mph. How fast are the two cars separating at the instant when the first car is 60miles north of A? {(14)(5(1/2))mph}

6) A balloon leaving the ground 60ft from an observer rises at the rate of 5ft/s. How fast

is the angle of elevation of the observer’s line of sight increasing at the instant when the balloon is at an altitude of 80ft? {3/100rads/s}

7) A searchlight S located on the ground 12m from a straight road is trained on a car

traveling on the road at a speed of 20m/s. Find the rate at which the searchlight is rotating at the instant when the car is 5m away from the point on the road nearest to the searchlight. {-240/169rads/s}

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