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Water Tank Problem
A water tank has the shape of an inverted circular cone with
a base radius of 2 meters and a height of 4 meters. If water
is being pumped into the tank at the rate of 2m3/min, find the
rate at which the water is rising when the water is 3 meters
deep.
2m
4m
r
h
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Sliding Ladder Problem
A ladder 10 feet long rests against a vertical wall. If the
bottom of the ladder slides away from the wall at a rate of 1
ft/sec, how fast is the top of the ladder sliding down the wall,
when the bottom of the ladder is 6 feet from the wall?
Wall
10 ft
y
Ground
x
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Highway Chase Problem
A police cruiser, approaching a right angled intersection from
the north, is chasing a speeding car that has turned the
corner and is now moving straight east. When the cruiser is
0.6mi north of the intersection and the car is 0.8mi to the
east, the police determine with radar that the distance
between them and the car is increasing at 20 mph. If the
cruiser is moving at 60mph at the instant of measurement,
what is the speed of the car?
N
Police
W
E
S
z
y
Speeder
x
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Rising Balloon Problem
A hot air balloon rising straight up from a level field is tracked
by a range finder 5oo ft from the lift off point. At the moment
the range finder’s elevation angle is  4 , the angle is
increasing at a rate of 0.14 radians/min. How fast is the
balloon rising at that moment?
h
Range
Finder
θ
500 ft
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Space Shuttle Problem
A television camera at ground level films the launch of the
space shuttle that is rising vertically according to the position
function s  50t 2 , where s is in feet and t is in seconds. The
camera is 2000 ft from the launch pad. Find the rate of
change in the angle of elevation of the camera 10 sec after
lift off.
Space
Shuttle
s
Camera
θ
2000 ft
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A spherical balloon is inflated with gas at a rate of 20
cubic feet per minute. How fast is the radius of the
balloon increasing at the instant the radius is:
a) 1 foot?
b) 2 feet?
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Hemispherical Reservoir Problem
Water is flowing at a rate of 6 m3/min from a reservoir
shaped like a hemispherical bowl of radius 13 m, shown
here in profile. The volume of a hemispherical bowl of radius
R is V 

3
y 2 3R  y  .
a. At what rate is the water level changing when the water
is 8 meters deep?
b. What is the radius of the water’s surface when the
water is y meters deep?
c. At what rate is the radius r changing when the water is
8 meters deep?
13
Water level
r
y
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Trough Problem
A trough is 12 feet long and 3 feet across the top. Its ends
are isosceles triangles with altitude of 3 feet.
a. If water is being pumped into the trough at 2 cubic
feet per minute, how fast is the water level rising
when h is 1 foot deep?
b. If the water is rising at a rate of 3 8 inch per minute,
when h  2 , determine the rate at which water is
being pumped into the trough?
2 ft3/min
12 ft
3 ft
h ft
3 ft
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Swimming Pool Problem
A swimming pool is 12 meters long, 6 meters wide and 1
meter deep at the shallow end, and 3 meters deep at the
deep end. Water is being pumped into the pool at ¼ cubic
meters per minute, and there is 1 meter of water at the deep
end.
a. What percent of the pool is filled?
b. At what rate is the water level rising?
¼ m3/min
1m
6m
3m
12 m