5128_Ch07_pp378-433

386 Chapter 7 Applications of Definite Integrals
Section 7.1 Exercises
In Exercises 1–8, the function v(t) is the velocity in msec of a
particle moving along the x-axis. Use analytic methods to do each of
the following:
(a) Determine when the particle is moving to the right, to the
left, and stopped.
(b) Find the particle’s displacement for the given time interval. If
s(0) 3, what is the particle’s final position?
(c) Find the total distance traveled by the particle.
1. vt 5 cos t, 0 t 2p See page 389.
2. vt 6 sin 3t, 0 t p2 See page 389.
3. vt 49 9.8t, 0 t 10 See page 389.
4. vt 6t 2 18t 12, 0 t 2 See page 389.
5. vt 5 sin 2 t cos t, 0 t 2p See page 389.
6. vt 4 t, 0 t 4 See page 389.
7. vt e sin t cos t, 0 t 2p See page 389.
t
8. vt , 0 t 3 See page 389.
1
t 2
9. An automobile accelerates from rest at 1 3t mphsec for
9 seconds.
(a) What is its velocity after 9 seconds? 63 mph
(b) How far does it travel in those 9 seconds?
344.52 feet
10. A particle travels with velocity
vt t 2 sin t msec
for 0 t 4 sec.
(a) What is the particle’s displacement? 1.44952 meters
(b) What is the total distance traveled?
1.91411 meters
11. Projectile Recall that the acceleration due to Earth’s gravity is
32 ftsec 2 . From ground level, a projectile is fired straight
upward with velocity 90 feet per second.
(a) What is its velocity after 3 seconds? 6 ft/sec
(b) When does it hit the ground?
5.625 sec
(c) When it hits the ground, what is the net distance it has
traveled? 0
(d) When it hits the ground, what is the total distance it has
traveled? 253.125 feet
In Exercises 12–16, a particle moves along the x-axis (units in cm).
Its initial position at t 0 sec is x0 15. The figure shows the
graph of the particle’s velocity vt. The numbers are the areas of
the enclosed regions.
4
5
a b c
24
12. What is the particle’s displacement between t 0 and t c?
23 cm
13. What is the total distance traveled by the particle in the same
time period? 33 cm
a:11 b:16 c: 8
14. Give the positions of the particle at times a, b, and c.
15. Approximately where does the particle achieve its greatest
positive acceleration on the interval 0, b? t a
16. Approximately where does the particle achieve its greatest
positive acceleration on the interval 0, c? t c
In Exercises 17–20, the graph of the velocity of a particle moving on
the x-axis is given. The particle starts at x 2 when t 0.
(a) Find where the particle is at the end of the trip.
(b) Find the total distance traveled by the particle.
17. v (m/sec)
(a) 6 (b) 4 meters
18.
19.
20.
2
0
1
0
–1
2
3
v (m/sec)
0 1
–3
v (m/sec)
0 1
–2
v (m/sec)
1 2 3 4
1 2 3 4
2 3 4 5 6 7
2 3 4 5 6 7
t (sec)
t (sec)
8 9 10
t (sec)
(a) 2
(a) 5
t (sec)
(b) 4 meters
(b) 7 meters
(a) 2.5
(b) 19.5 meters
21. U.S. Oil Consumption The rate of consumption of oil in the
United States during the 1980s (in billions of barrels per year)
is modeled by the function C 27.08 • e t25 , where t is the
number of years after January 1, 1980. Find the total consumption
of oil in the United States from January 1, 1980 to January 1,
1990. 332.965 billion barrels
22. Home Electricity Use The rate at which your home consumes
electricity is measured in kilowatts. If your home consumes
electricity at the rate of 1 kilowatt for 1 hour, you will be charged