related rates of change
example 1 cube
example 2 circle
example 3 sphere
example 4 triangle (sort of)
example 5 square
exersizes
1 Two people meet at the same point and start walking at 2 m/s at right-angles to each other,
marking out two sides of a square. At what rate is the area of the square increasing when
the two people are each 15 m from the origin?
2 An ink-blot is formed by dripping some ink continuously onto a sheet of blotting paper.
The radius of the blot is increasing at 0.6 mm/s. Calculate the rate at which the area is increasing:
(a) when the radius is 2 mm
(b) after 5 seconds.
3 The angle marked θ in this right-angled triangle is increasing at a rate of 1/15
radians/second. Calculate the rate at which the length marked x is increasing when θ = π/6.
marking out two sides of a square. At what rate is the area of the square increasing when
the two people are each 15 m from the origin?
2 An ink-blot is formed by dripping some ink continuously onto a sheet of blotting paper.
The radius of the blot is increasing at 0.6 mm/s. Calculate the rate at which the area is increasing:
(a) when the radius is 2 mm
(b) after 5 seconds.
3 The angle marked θ in this right-angled triangle is increasing at a rate of 1/15
radians/second. Calculate the rate at which the length marked x is increasing when θ = π/6.
4 The edge of a cube is decreasing at a rate of 5 mm/h. Calculate the rate at which the
surface area is decreasing when the edge is 40 cm.
5 A spherical balloon has a rate of increase of radius of 2 cm/s. Calculate the rate of
increase of:
(a) the volume
(b) the surface area
when the radius is 10 cm.
(Note: volume of sphere = 4πr^3/3 , (surface area of sphere = 4πr^2)
6 An ink-blot has an area which is increasing at a rate of 8 mm^2/s.
(a) At what rate is the radius increasing when the area is 30 mm^2?
(b) At what rate is the diameter increasing at this time?
7) A cube of ice is melting at a uniform rate. The initial volume of the cube is 20 cm^3, and the
volume after 5 minutes is 15 cm^3. Find the rate at which the edge of the cube is decreasing after 2 minutes.
8) A stationary tow-truck is using a winch to pull a car along the road towards the truck.
The winch is pulling the line in at 10 cm/s, and the top of the winch is 2.5 m above the ground.
How fast is the car moving towards the truck when it is 20 m away from the truck along the
road?
surface area is decreasing when the edge is 40 cm.
5 A spherical balloon has a rate of increase of radius of 2 cm/s. Calculate the rate of
increase of:
(a) the volume
(b) the surface area
when the radius is 10 cm.
(Note: volume of sphere = 4πr^3/3 , (surface area of sphere = 4πr^2)
6 An ink-blot has an area which is increasing at a rate of 8 mm^2/s.
(a) At what rate is the radius increasing when the area is 30 mm^2?
(b) At what rate is the diameter increasing at this time?
7) A cube of ice is melting at a uniform rate. The initial volume of the cube is 20 cm^3, and the
volume after 5 minutes is 15 cm^3. Find the rate at which the edge of the cube is decreasing after 2 minutes.
8) A stationary tow-truck is using a winch to pull a car along the road towards the truck.
The winch is pulling the line in at 10 cm/s, and the top of the winch is 2.5 m above the ground.
How fast is the car moving towards the truck when it is 20 m away from the truck along the
road?
solutions