5128_Ch07_pp378-433

426 Chapter 7 Applications of Definite Integrals
Group Activity In Exercises 13–16, the vertical end of a tank
containing water (blue shading) weighing 62.4 lbft 3 has the given
shape.
(a) Writing to Learn Explain how to approximate the force
against the end of the tank by a Riemann sum.
(b) Find the force as an integral and evaluate it.
13. semicircle (b) 1123.2 lb 14. semiellipse (b) 7987.2 lb
15. triangle (b) 3705 lb 16. parabola (b) 1506.1 lb
3
3 ft
8 ft
6 ft
6 ft
8 ft
4 ft 4.5 ft
17. (d) 1,494,240 ft-lb, 100 min; 1,500,000 ft-lb, 100 min
(d) The Weight of Water Because of differences in the
strength of Earth’s gravitational field, the weight of a cubic foot
of water at sea level can vary from as little as 62.26 lb at the
equator to as much as 62.59 lb near the poles, a variation of
about 0.5%. A cubic foot of water that weighs 62.4 lb in
Melbourne or New York City will weigh 62.5 lb in Juneau or
Stockholm. What are the answers to parts (a) and (b) in a
location where water weighs 62.26 lbft 3 ? 62.5 lbft 3 ?
18. Emptying a Tank A vertical right cylindrical tank measures
30 ft high and 20 ft in diameter. It is full of kerosene weighing
51.2 lbft 3 . How much work does it take to pump the kerosene
to the level of the top of the tank? 7,238,229 ft-lb
19. Writing to Learn The cylindrical tank shown here is to be
filled by pumping water from a lake 15 ft below the bottom
of the tank. There are two ways to go about this. One is to pump
the water through a hose attached to a valve in the bottom of the
tank. The other is to attach the hose to the rim of the tank and let
the water pour in. Which way will require less work? Give
reasons for your answer.
Through valve:
84,687.3 ft-lb
Over the rim:
98,801.8 ft-lb
Through a hose attached
to a valve in the bottom is
faster, because it takes
more time to do more
work.
2 ft
Open top
6 ft
6 ft
17. Pumping Water The rectangular tank shown here, with its top
at ground level, is used to catch runoff water. Assume that the
water weighs 62.4 lbft 3 .
Ground
level
10 ft
0 12 ft
Valve at base
20. Drinking a Milkshake The truncated conical container shown
here is full of strawberry milkshake that weighs 49 ozin 3 .
As you can see, the container is 7 in. deep, 2.5 in. across at the
base, and 3.5 in. across at the top (a standard size at Brigham’s
in Boston). The straw sticks up an inch above the top. About
how much work does it take to drink the milkshake through the
straw (neglecting friction)? Answer in inch-ounces.
91.3244 in.-oz
y
y
8 y
y
8
7
y 14x 17.5
20
y
Δy
y
(1.75, 7)
y 17.5
14
(a) How much work does it take to empty the tank by pumping
the water back to ground level once the tank is full? 1,497,600 ft-lb
(b) If the water is pumped to ground level with a
511-horsepower motor (work output 250 ft • lbsec),
how long will it take to empty the full tank (to the nearest
minute)? 100 min
(c) Show that the pump in part (b) will lower the water level
10 ft (halfway) during the first 25 min of pumping.
0
x
1.25
Dimensions in inches
21. Revisiting Example 3 How much work will it take to pump
the oil in Example 3 to a level 3 ft above the cone’s rim?
53,482.5 ft-lb