3.4 RELATED RATES
In this section we will study related rates problems. In such problems, one tries to find the rate
at which some quantity is changing by relating the quantity to other quantities whose rates of
change are known.
The phrase “the rate of change of 𝑣 with respect to 𝑟” means 𝑑𝑣 ⁄𝑑𝑟. If 𝑑𝑣 ⁄𝑑𝑟 is positive, then 𝑣
is increasing as 𝑟 increases. However, when 𝑑𝑣 ⁄𝑑𝑟 is negative then 𝑣 decreases as 𝑟 increases.
If we simply refer to the rate of change of 𝑣, then we mean 𝑑𝑣 ⁄𝑑𝑡 where 𝑡 represents time.
Example 1
A piece of paper is burning round the circumference of a circular hole. After 𝑡 seconds, the
radius, 𝑟 cm, of the whole is increasing at the rate of 0.5 𝑐𝑚⁄𝑠. Find the rate at which the area,
𝐴 𝑐𝑚 2 , of the hole is increasing when 𝑟 = 5 𝑐𝑚. 𝑎𝑛𝑠: 5𝜋 𝑐𝑚2 ⁄𝑠
Example 2
A ladder 5 m long is leaning against vertical wall. The bottom of the ladder is pulled along the
ground away from the wall at a constant rate of 0.2 𝑚𝑠 −1 . How fast will the top of the ladder
be falling at the instant it is 4 m above the ground?
𝑎𝑛𝑠: 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑎𝑑𝑑𝑒𝑟 𝑖𝑠 𝑓𝑎𝑙𝑙𝑖𝑛𝑔 𝑎𝑡 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 0.15 𝑚⁄𝑠
Example 3
Air is being pumped into a spherical balloon at the rate of 10 𝑐𝑚3 𝑠 −1. Find the rate of change
of the radius of the balloon when the radius of the balloon is 5 cm. 𝑎𝑛𝑠:
1
10𝜋
𝑐𝑚⁄𝑠
Example 4
Water is being poured into a conical vessel at a rate of 10 𝑐𝑚3 𝑠 −1. After 𝑡 seconds, the
volume, 𝑉 𝑐𝑚3, of water in the vessel is given by 𝑉 = 𝜋ℎ3 ⁄6, where ℎ is the depth of the
water, find in terms of ℎ, the rate at which the water is rising. 𝑎𝑛𝑠:
20
𝜋ℎ 2
𝑐𝑚⁄𝑠
Example 5
A particle moves along the curve 𝑦 = 𝑥 2 + 2𝑥. Find a point on the curve where both 𝑥 and 𝑦
1
3
coordinates of the particle change at the same rate. 𝑎𝑛𝑠: (− 2 , − 4)
3.4 Exercises
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
The area of a circle is increasing at the rate of 6𝜋 cm2 s−1 . Find the rate of increase of
the radius when the radius is 1.5 cm.
Water is poured into an inverted cone of depth 10 cm and radius 5 cm at a rate of 8
cm3s-1. Find the rate at which the depth of the water in the cone is increasing when
the depth is 4 cm.
The side of a cube is increasing at the rate of 6 cm s-1. Find the rate of increase of the
volume when the length of a side is 9 cm.
The radius of a circle is increasing at the rate of ½ cm s-1. Find the rate of increase of
the area when the radius is 4 cm.
The area of a square is increasing at the rate of 8 cm2 s−1. Find the rate of increase of
the length of a side when this area is 64 cm2.
The volume of a sphere is increasing at the rate of 16𝜋 cm3 s−1 . Find the rate of
increase of the radius when this radius is 8 cm.
The surface area of sphere is increasing at a rate of 4 cm2 s−1. Find the rate of
increase of the radius when the surface area is 64𝜋 cm2 .
A hollow right circular cone of height 18 cm and radius 12 cm is held vertex
downwards beneath a tap leaking at the rate of 2 cm s-1. Find the rate of rise of water
level when the depth is 6 cm.
A water tank in the shape of an inverted circular cone has radius of 20 m and a height
of 12 m. Water is pumped into the tank at a rate of
40 m3 /minute. Calculate the rate of change of the height of water in the tank when
the depth of water is 4 m.
Given a rectangle has length twice its breath. Find the rate of change of the perimeter
when the breadth of the rectangle is 1 m and its area is increasing at the rate of
18 cm2 s−1, assuming the expansion is uniform.
A hollow cone of base radius 20 cm and height 10 cm is held with its vertex
downwards. Water is poured into it at the rate of 15 cm3 s−1. Find the rate of increase
in the depth of the water when this depth is 3 cm. give your answer correct to 3
decimal places.
12.
13.
The above diagram shows a 10 m iron rod PQ with one end P resting on a vertical
wall and the other end resting on a horizontal ground at a distance 6 m from the foot
of the wall. The top end of the iron rod begins to slide down at a rate of 0.2 m s-1.
Calculate the rate of which the bottom end of this rod is moving away from the wall at
this instant.
1 1 1
Two variables 𝑢 and 𝑣 are connected by the relation + = ,
𝑢 𝑣 𝑓
where 𝑓 is a constant. Given that 𝑢 and 𝑣 are both vary with time 𝑡,
𝑑𝑢 𝑑𝑣
find an equation connecting
, , 𝑢 and 𝑣. Given also that 𝑣 is
𝑑𝑡 𝑑𝑡
decreasing at a rate of 2 cm per second, and that 𝑓 = 10 𝑐𝑚,
14.
15.
16.
17.
calculate the rate of increase of 𝑣 when 𝑢 = 50 cm.
The area of a circle is increasing at the rate of 12 cm2/s. Find the rate of increase of
the circumference when the radius is 3 cm.
The surface area of a cube is increasing at the rate of 3 cm/s. find the rate at which
the volume is increasing when the surface area is 6 cm2.
A baseball diamond is a square whose sides are 90 ft long. Suppose that a player
running from second base to third base has a speed of 30 ft/s at the instant when he
is 20 ft from third base. At what rate is the player’s distance from home plate
changing at that instant.
A camera mounted at a point 3,000 ft from the base of a rocket launching pad. If the
rocket is rising vertically at 880 ft/s when it is 4,000 ft above the launching pad, how
fast must the camera elevation angle change at that instant to keep the camera aimed
at the rocket?
3.4 Answers
1.
3.
5.
7.
9.
11.
13.
15.
17.
2 cm/s
1,458 cm3 s−1
0.5 cm s-1
1
cm⁄s
8𝜋
9
m⁄minute
10𝜋
0.133 cm s-1
1 𝑑𝑢 1 𝑑𝑣
+
= 0; 0.125 𝑐𝑚 𝑠 −1
𝑢2 𝑑𝑡 𝑣 2 𝑑𝑡
¾ cm3 s-1
66
𝑟𝑎𝑑/𝑠
625
10.
2⁄𝜋 cm2 s −1
2 cm2 s−1
1⁄16 cm⁄s
1
cm⁄s
8𝜋
0.27 cm s-1
12.
14.
0.15 m s-1
4 cm/s
2.
4.
6.
8.
16.
−
60
√85
𝑓𝑡/𝑠