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Worksheet: Related Rates

Q1:

If 𝑉 is the volume of a cube with edge length π‘₯ and the cube expands as time passes, give an expression for d d 𝑉 𝑑 .

  • A 3 π‘₯ 2
  • B π‘₯ π‘₯ 𝑑 2 d d
  • C π‘₯ π‘₯ 𝑑 3 d d
  • D 3 π‘₯ π‘₯ 𝑑 2 d d
  • E π‘₯ 2

Q2:

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 3 2 πœ‹ cm2/s with respect to time. Calculate the rate of increase of its radius when its base is 18 cm.

  • A 2 9 cm/s
  • B 8 2 7 cm/s
  • C 8 4 5 cm/s
  • D 4 2 7 cm/s

Q3:

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.

  • A cm2/cm
  • B 12 cm2/cm
  • C 6 cm2/cm
  • D cm2/cm

Q4:

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

Q5:

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.

Q6:

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.

  • A 7 1 5 rad/s
  • B 3 0 7 rad/s
  • C βˆ’ 1 4 1 5 rad/s
  • D βˆ’ 7 3 0 rad/s

Q7:

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.

Q8:

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 1 3 5 m/min
  • B 2 0 1 3 m/min
  • C 1 3 3 m/min
  • D 5 2 5 m/min

Q9:

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?

Q10:

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.

Q11:

A fine metal lamina is in the shape of a rectangle which has a length that is 3 5 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.

Q12:

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?

  • A 6 cm2/min
  • B 17 cm2/min
  • C 48 cm2/min
  • D βˆ’ 1 3 cm2/min

Q13:

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.

  • A 3 cm, 9 cm, 6 cm
  • B 2 cm, 8 cm, 4 cm
  • C 4 cm, 10 cm, 8 cm
  • D 1 cm, 7 cm, 2 cm

Q14:

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 3 6 1 3 cm/s
  • B √ 6 1 1 2 cm/s
  • C 5 4 cm/s
  • D 1 3 4 cm/s

Q15:

In a closed electric circuit, 𝑉 is the potential difference measured in volts, 𝐼 is the current intensity measured in amperes, and 𝑅 is the resistance measured in ohms. If the potential difference increases at a rate of 6 volts per second, and the current intensity decreases at a rate of 1 2 A/s, find the rate of change of the resistance when 𝑉 = 2 0 v o l t s and 𝐼 = 5 a m p e r e s .

  • A 5 4 Ξ©/s
  • B 4 5 Ξ©/s
  • C 5 8 Ξ©/s
  • D 8 5 Ξ©/s

Q16:

The volume of a known mass of gas, with a constant temperature, decreases at a constant rate 9 cm3/s. If its pressure is inversely proportional to the volume, and the pressure equals 2 200 g-wt/cm2 when the volume is 250 cm3, find the rate of change of the pressure with respect to time when the volume of gas is 100 cm3.

  • A 3 9 6 5 g-wt/cm2/s
  • B 49 500 g-wt/cm2/s
  • C 198 g-wt/cm2/s
  • D 495 g-wt/cm2/s

Q17:

A point is moving on the curve of the function 𝑓 ( π‘₯ ) = √ π‘₯ + 2 2 . If its π‘₯ -coordinate increases at a rate of 9 √ 1 5 cm/s, find the rate of change of the distance between this point and the point ( 1 , 0 ) at π‘₯ = 3 .

Q18:

Let 𝐴 be the area between concentric circles of radii π‘Ÿ 1 and π‘Ÿ 2 , where π‘Ÿ > π‘Ÿ 2 1 . When π‘Ÿ = 1 1 c m and is increasing at 0.5 cm/s, π‘Ÿ = 5 2 c m and is decreasing at 0.1 cm/s. What is the rate of change of 𝐴 at this moment?

  • A 2 πœ‹ cm2/s
  • B 0 cm2/s
  • C βˆ’ 5 πœ‹ cm2/s
  • D βˆ’ 2 πœ‹ cm2/s

Q19:

Suppose 𝑦 = √ 4 π‘₯ βˆ’ 3 , where π‘₯ and 𝑦 are functions of 𝑑 . If d d π‘₯ 𝑑 = 5 , find d d 𝑦 𝑑 when π‘₯ = 3 .

  • A 2 0 3
  • B 2 3
  • C 2 1 5
  • D 1 0 3
  • E 5 6

Q20:

Suppose that functions π‘₯ and 𝑦 of 𝑑 satisfy 4 π‘₯ + 9 𝑦 = 3 6 2 2 . When π‘₯ = 1 and 𝑦 = 4 √ 2 9 , we have d d 𝑦 𝑑 = 1 2 . What is d d π‘₯ 𝑑 ?

  • A √ 2 2
  • B βˆ’ √ 2
  • C βˆ’ 2 √ 2
  • D βˆ’ √ 2 2
  • E βˆ’ 9 8

Q21:

The internal radius of a hollow sphere is increasing at the rate of 5.3 cm/s, while the volume of the material of the sphere is constant. When its radii are 5 cm and 6 cm, find the rate of change of its thickness approximated to the nearest hundredth.

Q22:

A particle moves along the hyperbola . If at the point , what is ?

  • A
  • B
  • C
  • D
  • E

Q23:

The radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 3 cm?

  • A 1 2 πœ‹ cm2/min
  • B 7 2 πœ‹ cm2/min
  • C 2 4 πœ‹ cm2/min
  • D 4 8 πœ‹ cm2/min
  • E 3 6 πœ‹ cm2/min

Q24:

The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the sphere’s diameter is 20 mm?

  • A 2 0 0 πœ‹ mm3/s
  • B 1 6 0 πœ‹ mm3/s
  • C 4 0 0 πœ‹ mm3/s
  • D 8 0 0 πœ‹ mm3/s
  • E 8 0 πœ‹ mm3/s

Q25:

A triangle with sides π‘Ž , 𝑏 , and contained angle πœƒ has area 𝐴 = 1 2 π‘Ž 𝑏 πœƒ s i n . Suppose that π‘Ž = 4 , 𝑏 = 5 , and the angle πœƒ is increasing at 0.6 rad/s. How fast is the area changing when πœƒ = πœ‹ 3 ?