Worksheet: Related Rates

In this worksheet, we will practice using derivatives to find the relation between the rates of two or more quantities in related rates problems.


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

  • A3π‘₯
  • Bπ‘₯
  • C3π‘₯π‘₯π‘‘οŠ¨dd
  • Dπ‘₯π‘₯π‘‘οŠ¨dd
  • Eπ‘₯π‘₯π‘‘οŠ©dd


A triangle with sides π‘Ž, 𝑏, and contained angle πœƒ has area 𝐴=12π‘Žπ‘πœƒsin. 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?


Given that a rocket of mass 26 metric tons is burning fuel at a constant rate of 80 kg/s, find the mass of the rocket 25 seconds after take-off.


A rocket, coming to the end of its journey, is consuming its fuel at a rate of 80 kg/s. Given that its mass when empty is 7 metric tons and that its current mass is 9 metric tons, how long will it take for it to consume its remaining fuel?


The area of a circular disk is increasing at 15 cm2/s. What is the rate of increase of its radius when the radius is 6 cm? Use πœ‹=227 to simplify your answer.

  • A7660 cm/s
  • Bβˆ’71,320 cm/s
  • C7220 cm/s
  • D71,320 cm/s


A particle is moving along the curve 6𝑦+2π‘₯βˆ’2π‘₯+5π‘¦βˆ’13=0. If the rate of change of its π‘₯-coordinate with respect to time as it passes through the point (βˆ’1,3) is 2, find the rate of change of its 𝑦-coordinate with respect to time at the same point.

  • A823
  • B1241
  • C136
  • Dβˆ’207


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?

  • A9641 cm2/s
  • B4841 cm2/s
  • C1241 cm2/s
  • Dβˆ’4841 cm2/s


A body moves along the curve 𝑦=5π‘₯. If ddπ‘₯𝑑=βˆ’3 units/second at 𝑦=βˆ’5, what is dd𝑦𝑑 at this moment?

  • Aβˆ’32 units/second
  • B32 units/second
  • C3 units/second
  • Dβˆ’12 units/second


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 85πœ‹ cm2/min. Determine the radius of the circle at that moment.


The height of a cylinder is equal to its base diameter. Maintaining this relationship between the height and base diameter, the cylinder expands such that the rate of increase of its surface area is 32πœ‹ cm2/s with respect to time. Calculate the rate of increase of its radius when its base has a radius of 18 cm.

  • A427 cm/s
  • B845 cm/s
  • C827 cm/s
  • D29 cm/s


A lamina is shaped like an isosceles trapezoid. Its parallel sides are π‘₯ cm and 3π‘₯ cm and one of its vertices measures 45∘. The lamina shrinks while preserving its shape. Determine the rate of change of its area with respect to π‘₯ when its height is 3 cm.

  • A12 cm2/cm
  • Bβˆ’12 cm2/cm
  • Cβˆ’6 cm2/cm
  • D6 cm2/cm


A water pipe, denoted by 𝐴𝐡, has a length of 5 meters and is placed with its end 𝐴 on a horizontal ground, leaning against the edge of a vertical wall of length 3 meters at point 𝐷. Given that end 𝐴 slides away from the wall at a rate of 85 m/min, determine the rate at which end 𝐡 slides till it reaches the ground.

  • A12125 m/min
  • Bβˆ’96125 m/min
  • C96125 m/min
  • Dβˆ’192125 m/min


𝐴𝐢 and οƒͺ𝐡𝐢 are two orthogonal roads, where 𝐴𝐢=13m and 𝐡𝐢=66m. A person started walking from 𝐴 toward 𝐢 with a uniform velocity of 1 m/s, and another person started walking from 𝐡 toward 𝐢 with a uniform velocity of 7 m/s. Find the rate of change of the distance between them after 6 seconds.


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.

  • A307 rad/s
  • Bβˆ’730 rad/s
  • C715 rad/s
  • Dβˆ’1415 rad/s


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.


A man pulls a car toward 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?

  • A2013 m/min
  • B135 m/min
  • C133 m/min
  • D525 m/min


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?


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.


A fine metal lamina is in the shape of a rectangle which has a length that is 35 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.


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
  • B17 cm2/min
  • C6 cm2/min
  • Dβˆ’13 cm2/min


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.

  • A2 cm, 8 cm, 4 cm
  • B1 cm, 7 cm, 2 cm
  • C3 cm, 9 cm, 6 cm
  • D4 cm, 10 cm, 8 cm


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.

  • A3613 cm/s
  • B134 cm/s
  • C√6112 cm/s
  • D54 cm/s


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 12 A/s, find the rate of change of the resistance when 𝑉=20volts and 𝐼=5amperes.

  • A85 Ξ©/s
  • B45 Ξ©/s
  • C58 Ξ©/s
  • D54 Ξ©/s


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.

  • A198 g-wt/cm2/s
  • B49,500 g-wt/cm2/s
  • C3965 g-wt/cm2/s
  • D495 g-wt/cm2/s


A point is moving on the curve of the function 𝑓(π‘₯)=√π‘₯+2. If its π‘₯-coordinate increases at a rate of 9√15 cm/s, find the rate of change of the distance between this point and the point (1,0) at π‘₯=3.

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