Related Rate Problems - Solutions
1.
Air is being pumped into a spherical balloon so that its volume increases at a
rate of 100 cm3/s. How fast is the radius of the balloon increasing when the
diameter is 50 cm?
4
3
V= r
3
dV
2 dr
=4  r
dt
dt
100=4 25
2
dr
dt
dr
1
=
cm/s.
dt 25
2. A ladder 10ft long rests against a vertical wall. If the bottom of the ladder
slides away from the wall at a rate of 1 ft/s. how fast is the top of the ladder
sliding down the wall when the bottom of the ladder is 6 ft from the wall?
x 2 y 2=100
y
dx
dy
2 y =0
dt
dt
10
2x
x
26128
dy
3
=− ft/s.
dt
4
dy
=0
dt
3. A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks
away from the pole with a speed of 5 ft/s along a straight path. How fast is
the tip of his shadow moving when he is 40 ft from the pole?
15
6
x
y
By similar triangles,
15
6
=
. That is, 9 y=15 x .
y y −x
9
dy
dx
=15
dt
dt
9
dy
=155
dt
dy 25
=
ft/s.
dt 3
4. Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h.
Both are headed for the intersection of the two roads. At what rate are the
cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from
the intersection?
x
y
x 2 y 2=z 2
2x
z
dx
dy
dz
2 y =2 z
dt
dt
dt
20.3−5020.4 −60=20.5 
dz
=−78 mi/h.
dt
dz
dt
5.
A water tank has the shape of an inverted circular cone with base radius 2m
and height 4m. If water is being pumped into the tank at a rate of 2 m3/min,
find the rate at which the water level is rising when the water is 3 m deep.
2
1
V =  r2 h
3
4
r
by similar triangles
h
V=
r 2
h
= and hence r=
h 4
2
1
3
h
12
dV 1
dh
= 3 h2
dt 12
dt
2=
1
dh
3 32
12
dt
dh 8
=
m/min.
dt 9 
6.
A hot air balloon rising straight up from a level field is being tracked from a
spectator 500 ft from the liftoff point. At the moment the angle of elevation is
π
, the angle is increasing at the rate of 0.14 rad/min. How fast is the
4
balloon rising at that moment?
tan =
y
2
Ө
sec 
500
sec
2
y
500
d
1 dy
=
dt 500 dt
 4 0.14 = 5001 dydt
dy
=140 ft/min.
dt
7. A mechanic is reboring a 6-in-deep cylinder to fit a new piston. The machine
they are using increases the cylinder’s radius one-three thousandth of an
inch every minute. How rapidly is the cylinder volume increasing when the
bore diameter is 3.8 inches?
2
V = r 6
r
dV
dr
=12 r
dt
dt
6
dV
1
=121.9
dt
3000
dr 19 
=
in3/min
dt 2500