RELATED RATES PROBLEMS
1. A raindrop falls in a puddle and the ripples spread in circles, the radii
of which grow at the rate of 2 cm/s. Find the rate of increase of the area
of the circle when the radius is 6 cm.
𝑚3
2. A spherical balloon is being inflated at the rate of 3𝜋
. Twelve
𝑚𝑖𝑛
minutes after the inflation first begins, what is the rate of increase of the
diameter of the balloon with respect to time?
3. A spherical balloon is inflated such that its radius is increasing at the
𝑐𝑚
rate of 1
. Find the rate of increase of the volume
𝑚𝑖𝑛
(a) when the diameter is 2 m?
(b) when the surface area is 324𝜋 𝑐𝑚2 ?
4. A weather balloon with radius 9 m springs a leak, losing air at
171 𝜋
𝑚3
𝑚𝑖𝑛
. Find the rate of decrease of the radius after 4 min.
5. A weather balloon rises through the air at the rate of 500 m/min.
Every 1000 m the decrease of air pressure outside the balloon causes its
radius to increase by 8 cm. How rapidly is the volume increasing at the
instant its radius is 90 cm?
6. A cylindrical tank has a radius of 3 m and a depth of 10 m. It is being
filled at the rate of 5
𝑚3
𝑚𝑖𝑛
. How fast is the surface rising?
7. A rectangular tank has the following dimensions: length is 10 m,
width is 6 m, and depth is 8 m. It is being filled with water and the
𝑐𝑚
surface level is rising at 6
. What is the rate of flow of water into the
𝑚𝑖𝑛
tank?
8. An inverted conical tank has a total depth 2 m, and the radius of the
top is 0.8 m. If water is running out of the tank at the rate of 2
fast is the level of water decscending at the following times?
a) when the depth of the water is 0.5 m
b) when the tank is half full
𝑚3
𝑚𝑖𝑛
, how
9. A conical flow vase is 30 cm high with a radius of 6 cm at the top. If it
𝑐𝑚3
is being filled with water at the rate of 10
, find the rate at which the
𝑠
water level is rising when the depth is 20 cm.
10. Sand pouring from a conveyor belt forms a conical pile such that the
height h is twice the radius r. If the sand is pouring from the belt at the
rate of 8𝜋
begins?
𝑚3
𝑚𝑖𝑛
, how fast is the height increasing 18 min after pouring
𝑐𝑚3
11. A conical glass vase is being filled with a liquid at the rate of 10
.
𝑠
The vase is 20 cm high and 3 cm in radius at the top. Find the rate at
which the water level is rising when the depth is 10 cm.
12. A horizontal eavestrough 3 m long has a triangular cross section 10
cm across the top and 10 cm deep. During a rainstorm the water in the
𝑐𝑚
trough is rising at the rate of 1
when the depth is 5 cm.
𝑚𝑖𝑛
a) How fast is the volume of water in the trough increasing?
b) After the rain stopped, the water drained out of the trough at the rate
𝑚3
of 0.06
. How fast is the surface of the water falling when the depth
𝑚𝑖𝑛
is 1 cm?
13. When air expands adiabatically, the pressure P and volume V satisfy
the equation 𝑃𝑉1.4 = 𝑘 (a constant). At a certain instant the volume is
100 𝑐𝑚3 and is decreasing at the rate of 10
of the pressure in terms of k.
𝑐𝑚3
𝑠
. Find the rate of change
14. A kite flying 100 m high is blown horizontally by the wind at the
𝑚
velocity of 4 . If the string is played out from a fixed position, how fast
𝑠
is the length of the string increases when it is 125 m long?
15. Ship A approaches a harbour entrance from the nirth at 15 kn, (knot
= one nautical mile per hour), while ship B approaches from the west at
18 kn. How rapidly are the ships approaching each other at the instant
both are 5 nmi (nautical mile) from the entrance?
16. A plane flying north at 600 km/h passes over a town at 12:00
exactly. A second plane flying east at 540 km/h passes over the town at
12:01. If the altitudes of the two aitcrafts are the same, how fast are they
moving apart at 12:06?
17. A ladder 10 m long leans against a vertical wall. If the bottom slides
out at a rate of 1 m/min, how fast is the top descending when the bottom
is 6 m away from the wall?
18. A pedestrain 2 m tall waks directly away from a srteet light 6 m
𝑚
above the ground at 80
. Determine the following at the instant he is
𝑚𝑖𝑛
8 m from the base of the light post:
a) the velocity of the end of his shadow;
b) the rate of increase in the length of his shadow.
Answers:
𝑐𝑚2
1. 24 𝜋
𝑠
2. 2. 0.16
𝑚
𝑚𝑖𝑛
3. a) 40 000 𝜋
𝑚
4. −1.19
5
𝑚
9
𝑚𝑖𝑛
𝑚3
7. 3.6
9.
8
𝑚
𝑠
10. 0.22
11.
40
9
𝑚𝑖𝑛
𝑐𝑚
𝜋
𝜋
b) 324 𝜋
𝑚𝑖𝑛
𝑚𝑖𝑛
8. a) −15.9
5
𝑚𝑖𝑛
𝑚𝑖𝑛
𝑚3
5. 0.1296 𝜋
6. 𝜋
𝑐𝑚3
b) 3.98
𝑚
𝑚𝑖𝑛
𝑐𝑚
𝑠
𝑐𝑚3
𝑚
12. a) 1500
b) 2
𝑚𝑖𝑛
𝑚𝑖𝑛
13. 0.00022k
𝑚
14. 2.4
15.
𝑠
33√2
2
16. 804
𝑘𝑛
𝑘𝑚
ℎ
17. −0.75
18. a) 120
𝑚
𝑚𝑖𝑛
𝑚
𝑚𝑖𝑛
b) 40
𝑚
𝑚𝑖𝑛
𝑚
𝑚𝑖𝑛
𝑐𝑚3
𝑚𝑖𝑛