Workshop #18
Professor D. Olles
1. Air is being pumped into a spherical balloon at a rate of 5 cubic centimeters per minute. Determine the rate at which the radius of the balloon is
increasing when the diameter of the balloon is 20 cm.
2. A 15 foot ladder is resting against the wall. The bottom is 10 feet away
from the wall and is being pushed towards the wall at a rate of 1/4 feet
per second. How fast is the top of the ladder moving up the wall?
3. Two people are 50 feet apart. One of them starts walking north at a rate
so that the angle shown in the diagram below is changing at a constant
rate of 0.01 radians per minute. At what rate is distance between the two
people changing when the angle is π/3 radians?
1
4. A tank of water in the shape of a cone is leaking water at a constant rate
of 2 cubic feet per hour. The base radius of the tank is 5 feet and the
height of the tank is 14 feet.
a. At what rate is the depth of the water in the tank changing when
the depth of the water is 6 ft?
b. At what rate is the radius of the top of the water in the tank
changing when the depth of the water is 6 ft?
5. A trough of water is 8 meters deep and its ends are in the shape of isosceles
triangles whose width is 5 meters and height is 2 meters. If water is being
pumped in at a constant rate of 6 cubic meters per second. At what rate
is the height of the water changing when the water has a height of 120
cm?
2
Solutions
1. Air is being pumped into a spherical balloon at a rate of 5 cubic centimeters per minute. Determine the rate at which the radius of the balloon is
increasing when the diameter of the balloon is 20 cm.
dV
=5
dt
r = 10
dr
=?
dt
4
V = πr3
3
dV
4
dr
dr
= π(3r2 )
= 4πr2
dt
3
dt
dt
dr
5 = 4π(10)2
dt
dr
5 = 4π(100)
dt
dr
5 = 400π
dt
5
dr
=
400π
dt
dr
1
=
cm/min
dt
80π
3
2. A 15 foot ladder is resting against the wall. The bottom is 10 feet away
from the wall and is being pushed towards the wall at a rate of 1/4
feet per second. How fast is the top of the ladder moving up the wall?
x2 + y 2 = z 2
102 + y 2 = 152
100 + y 2 = 225
y 2 = 125
√
y=5 5
x2 + y 2 = z 2
dx
dy
dz
+ 2y
= 2z
dt
dt
dt
√
−1
dy
= 2(15)(0)
+ 2(5 5)
2(10)
4
dt
√ dy
−5 + 10 5
=0
dt
√ dy
10 5
=5
dt
√
dy
1
5
=− √ =
f t/sec
dt
10
2 5
2x
4
3. Two people are 50 feet apart. One of them starts walking north at a rate
so that the angle shown in the diagram below is changing at a constant
rate of 0.01 radians per minute. At what rate is distance between the two
people changing when the angle is π/3 radians?
π
3
dθ
1
=
dt
100
x =?
dx
=?
dt
50
cos θ =
x
π 50
=
cos
3
x
1
50
=
2
x
x = 100
50
cos θ =
x
dθ
dx
− sin θ
= −50x−2
dt
dt
π 1
50 dx
sin
=
3 100
1002 dx
√
3
1 dx
=
2
2 dt
dx √
= 3f t/min
dt
θ=
5
4. A tank of water in the shape of a cone is leaking water at a constant rate
of 2 cubic feet per hour. The base radius of the tank is 5 feet and the
height of the tank is 14 feet.
dV
= −2
dt
r
h
=
5
14
5h
r=
14
2
1
1
1
5h
25h2
25π 3
V = πr2 h = π
h= π
h=
h
3
3
14
3
196
588
a. At what rate is the depth of the water in the tank changing when
the depth of the water is 6 ft?
25π 2 dh
25π 2 dh
dV
=
3h
=
h
dt
588
dt
196 dt
25π 2 dh
(6 )
196
dt
25π
dh
−2 =
(36)
196
dt
25π
dh
−2 =
(9)
49
dt
49
dh
= −2
dt
225π
−2 =
dh
98
=−
f t/hr
dt
225π
b. At what rate is the radius of the top of the water in the tank
changing when the depth of the water is 6 ft?
r=
5h
14
dr
5 dh
=
dt
14 dt
dr
5
98
=
−
dt
14
225π
dr
1
49
=
−
dt
7
45π
dr
7
=−
f t/hr
dt
45π
6
5. A trough of water is 8 meters deep and its ends are in the shape of isosceles
triangles whose width is 5 meters and height is 2 meters. If water is being
pumped in at a constant rate of 6 cubic meters per second. At what rate
is the height of the water changing when the water has a height of 120
cm?
dV
=6
dt
6
h = 120cm = m
5
dh
=?
dt
1
V = wh(8) = 4wh
2
w
h
=
5
2
5h
w=
2
5h
V =4
h
2
V = 10h2
dV
dh
= 20h
dt
dt
6 dh
6 = 20
5 dt
dh
1
= m/sec
dt
4
7