# Worksheet: Related Rates

Q1:

If is the volume of a cube with edge length and the cube expands as time passes, give an expression for .

• A
• B
• C
• D
• E

Q2:

A particle is moving along the curve . If the rate of change of its -coordinate with respect to time as it passes through the point is 2, find the rate of change of its -coordinate with respect to time at the same point.

• A
• B
• C
• D

Q3:

A body moves along the curve . If at , what is at this moment?

• A
• B
• C
• D

Q4:

A spherical balloon leaks helium at the rate of 48 cm3/s. What is the rate of change of its surface area when its radius is 41 cm?

• A cm2/s
• B cm2/s
• C cm2/s
• D cm2/s

Q5:

At a certain moment, the radius of a circle increases at a rate of 17 cm/min, and its area increases at a rate of cm2/min. Determine the radius of the circle at that moment.

Q6:

At 8 am, a ship sailed at 14 km/h westward to a port 53 km away. At 9 am, another ship left the port sailing south at 60 km/h. What was the rate of change of the distance between the ships at 10 am?

Q7:

The height of a cylinder is equal to its base diameter. The cylinder expands such that the rate of increase of its surface area is cm2/s with respect to time. Calculate the rate of increase of its radius when its base is 18 cm.

• A cm/s
• B cm/s
• C cm/s
• D cm/s

Q8:

A lamina is shaped like an isosceles trapezoid. Its parallel sides are cm and cm and one of its vertices measures . The lamina shrinks while preserving its shape. Determine the rate of change of its area with respect to when its height is 3 cm.

• A6 cm2/cm
• B cm2/cm
• C cm2/cm
• D12 cm2/cm

Q9:

A trough is 8 ft long and its ends have the shape of isosceles triangles that are 2 ft across the top and have a height of 4 ft. Suppose the volume of a prism of height and base of area is given by . If the trough is being filled with water at a rate of 9 ft3/min, how fast is the water level rising when the water is 3 inches deep?

Q10:

A water pipe, denoted by , has a length of 5 meters and is placed with its end on a horizontal ground, leaning against the edge of a vertical wall of length 3 meters at point . Given that end slides away from the wall at a rate of m/min, determine the rate at which end slides till it reaches the ground.

• A m/min
• B m/min
• C m/min
• D m/min

Q11:

A runner sprints around a circular track of radius 100 m at a constant speed of 8 m/s. The runner’s friend is standing at a distance 250 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 250 m? Give your answer to two decimal places.

Q12:

A man starts walking north at 180 ft/min from a point P. Five minutes later, a woman starts walking south at 300 ft/min from a point 466 ft due east of P. How fast are they walking away from each other twelve minutes after the woman starts walking? Round your result to two decimal places.

Q13:

and are two orthogonal roads, where and . A person started walking from towards with a uniform velocity of 1 m/s, and another person started walking from towards with a uniform velocity of 8 m/s. Find the rate of change of the distance between them after 7 seconds.

Q14:

A television camera is positioned 1 336 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Suppose the rocket rises vertically and its speed is 501 ft/s when it has risen 1 002 ft. If the television camera is always kept aimed at the rocket, how fast is the camera’s angle of elevation changing at that same moment?

Q15:

At the end of a 500-meter race, a runner ran in a straight line at 3.5 m/s. A camera was positioned 3 meters from the finish line such that it was perpendicular to the track and on the same horizontal plane as the runner. Determine the rate of change of the angle by which the camera rotated to capture the runner when he was 6 meters away from the finish line.

Q16:

The length of a rectangle is increasing at a rate of 15 cm/s and its width at a rate of 13 cm/s. Determine the rate at which the area of the rectangle increases when the length of the rectangle is 25 cm and its width is 12 cm.

Q17:

A man pulls a car towards himself with the use of a rope on a pulley attached 24 m above the ground, directly above him. If he pulls the rope at 4 m/min, what is the speed of the car when it is 10 m away from him?

• A m/min
• B m/min
• C m/min
• D m/min

Q18:

Two ships leave a port at the same time. The first sails east at 30 km/h. The other sails south at 40 km/h. What is the rate of change of the distance between them 5 hours later?

Q19:

A rectangular piece of metal has a length that exceeds its width by 45 cm, and upon cooling, it shrinks in a way that its length is always 45 cm more than its width. Given that its length decreases at a rate of 0.026 cm/s at the moment when its width is 90 cm, determine the rate of change of its area at this moment.

Q20:

A fine metal lamina is in the shape of a rectangle which has a length that is of that of its diagonal. The lamina shrinks by cooling while preserving its geometrical shape and the same ratio between its dimensions. At a certain moment, the length of its diagonal and its surface area decrease at a rate of 1.5 cm/min and 36 cm2/min, respectively. Determine the surface area of the lamina at that moment.

Q21:

The inner radius of a hollow sphere is increasing at a rate of 1.5 cm/s, while the volume of the sphere’s material remains constant. Determine the rate of change of its outer radius when its radii are 8 cm and 14 cm.

• A0 cm/s
• B0.49 cm/s
• C0.86 cm/s
• D cm/s

Q22:

The sides adjacent to the right angle of a triangle are initially of lengths 7 cm and 1 cm. The first side then decreases at the rate of 1 cm/min while the second side increases at the rate of 5 cm/min. At what rate is the area of the triangle changing 6 minutes later?

• A48 cm2/min
• B cm2/min
• C6 cm2/min
• D17 cm2/min

Q23:

A rectangular box expands while maintaining its relative dimensions. Its length is 6 cm more than its width. Its height is 2 times its width. Find the box’s dimensions at the moment when its volume is increasing at 0.3 cm3/min and its width is increasing at 0.01 cm/min.

• A4 cm, 10 cm, 8 cm
• B1 cm, 7 cm, 2 cm
• C3 cm, 9 cm, 6 cm
• D2 cm, 8 cm, 4 cm

Q24:

A right circular cylinder-like container has an interior height of 5 cm and an interior base radius of 6 cm. A metal rod 16 cm long is placed inside the container. If the rate at which the rod slides away from the container’s edge is 3 cm/s, determine the rate at which the rod slides on the container’s base the moment it touches the container’s wall.

• A cm/s
• B cm/s
• C cm/s
• D cm/s