Application Chapter 4
1.
Gas is escaping from a spherical balloon at the rate of 2 ft 3 / min . How fast is the
surface area changing when the radius is 12 ft ?
2.
A train, starting at 11am, travels east at 45mph while another, starting at noon from
the same point, travels south at 60mph . How fast are they separating at 3pm?
3.
Water is being withdrawn from a conical reservoir 3 feet in radius and 10 feet deep at
a rate of 4 ft 3 / min . How fast is the surface falling when the depth of the water is 6 feet?
How fast is the area of the surface decreasing at this instant?
4.
A ladder 13 feet long is leaning against the side of a building. If the foot of the ladder
is pulled away from the building at a constant rate of 2 inches per second, how fast is the
angle formed by the ladder and the ground changing (in radians per second) at the instant
when the top of the ladder is 12 feet above the ground?
5.
A conical paper cup 3 inches across the top and 4 inches deep is full of water. The cup
springs a leak at the bottom and loses water at the rate of 2 cubic inches per minute. How fast
is the water level dropping at the instant when the water is exactly 3 inches deep?
6.
What is the area bounded by the curves y  x 2  1 and y  2 x  7 ?
7.
Find the area of a region enclosed by the parabola y 
1 2
x and the two lines y  x
9
and y  1using the integration with respect to x .
8.
Find the exact x and y coordinates of the stationary points and inflexion points of the
following functions:
i)
f ( x)  x3  12 x  5
ii)
f ( x)  3 x 4  4 x 3  1
iii)
f ( x)  x 3  2 x 2  x
iv)
f ( x)  x 4  2 x 3  6
Answer:
1.
4
dS
1
V   r 3 , S  4 r 2 
  ft 2 / min
3
dt
3
2.
74.25mph
3.
1
V   r 2h
3
A   r2
dh
100

ft / min
dt
81
dA
4
  ft 2 / min
dt
3
4.
dA 119 2

ft / sec
dt
36
5.
dh
128

in / min
dt
81
6.
A  36unit 2
7.

8.
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