5128_Ch07_pp378-433

Section 7.5 Applications from Science and Statistics 429
47. Football A quarterback threw a 14.5-oz football 88 ftsec
(60 mph). How many foot-pounds of work were done on the
ball to get it to that speed? 109.7 ft-lb
48. Softball How much work has to be performed on a
6.5-oz softball to pitch it at 132 ftsec (90 mph)? 110.6 ft-lb
42. F m d v
mv d v
, so W x 2
F(x) dx
dt
dx
x 1
x 2
mv d v
dx
x 1
v 2
mv dv 1 dx
v 1
2 mv 2 2 1 2 mv 1 2
Quick Quiz for AP* Preparation: Sections 7.4 and 7.5
49. A Ball Bearing A 2-oz steel ball bearing is placed on a vertical
spring whose force constant is k 18 lbft. The spring is
compressed 3 in. and released. About how high does the ball
bearing go? (Hint: The kinetic (compression) energy, mgh, of a
spring is 1 2 ks 2 , where s is the distance the spring is compressed,
m is the mass, g is the acceleration of gravity, and h is the height.)
4.5 ft
You should solve the following problems without using a
4. Free Response The front of a fish tank is rectangular in shape
graphing calculator.
and measures 2 ft wide by 1.5 ft tall. The water in the tank exerts
1. Multiple Choice The length of a curve from x 0 to x 1 is pressure on the front of the tank. The pressure at any point on the
given by 1 front of the tank depends only on how far below the surface the
1 16
0 x6 dx. If the curve contains the point (1, 4),
point lies and is given by the equation p 62.4h, where h is
which of the following could be an equation for this curve? A
depth below the surface measured in feet and p is pressure
(A) y x 4 3
measured in pounds/ft 2 .
(B) y x 4 1
(C) y 1 16x 6
(D) y 1 x 16
6
1.5 ft
(E) y x x h
7
7
h
2. Multiple Choice Which of the following gives the length of
the path described by the parametric equations x 1 4 t 4 and
2 ft
y t 3 , where 0 t 2? D
The front of the tank can be partitioned into narrow horizontal
2
(A) t 6 9t 4 bands of height Δh. The force exerted by the water on a band at
dt
depth h
0
i is approximately
2
pressure area = 62.4h
(B)
i 2Δh.
t 6 1 dt
0
(a) Write the Riemann sum that approximates the force exerted
2
on the entire front of the tank.
(C) 1 9t 4 dt
(b) Use the Riemann sum from part (a) to write and evaluate a
0
definite integral that gives the force exerted on the front of the
2
tank. Include correct units.
(D) tt 6 9 4 dt
0
(c) Find the total force exerted on the front of the tank if the front
2
(and back) are semicircles with diameter 2 ft. Include correct units.
(E) t 3 3t 2 dt
0
2 ft
3. Multiple Choice The base of a solid is a circle of radius
2 inches. Each cross section perpendicular to a certain diameter
is a square with one side lying in the circle. The volume of the (a) n
62.4h i 2 h
solid in cubic inches is C
i1
(A) 16 (B) 16p (C) 12 8
(D) 12 8p (E) 32p (b) 1.5
62.4h 2 dh 140.4 lbs (c) 1.5
62.4h 21h 2 dh 41.6 lbs
3 3
0
0