5128_Ch03_pp098-184

35. Exit velocity 348.712 ft/sec 237.758 mi/h
Figure 3.34 Two balls falling from rest. (Exercise 38)
33. Pisa by Parachute (continuation of Exercise 18) A few
years ago, Mike McCarthy parachuted 179 ft from the top of the
Tower of Pisa. Make a rough sketch to show the shape of the
graph of his downward velocity during the jump.
34. Inflating a Balloon The volume V 43pr 3 of a
spherical balloon changes with the radius.
(a) At what rate does the volume change with respect to the
radius when r 2 ft? 16 cubic feet of volume per foot of radius
(b) By approximately how much does the volume increase when
the radius changes from 2 to 2.2 ft? By about 11.092 cubic feet
35. Volcanic Lava Fountains Although the November 1959
Kilauea Iki eruption on the island of Hawaii began with a line
of fountains along the wall of the crater, activity was later
confined to a single vent in the crater’s floor, which at one point
shot lava 1900 ft straight into the air (a world record). What was
the lava’s exit velocity in feet per second? in miles per hour?
[Hint: If v 0 is the exit velocity of a particle of lava, its height t
seconds later will be s v 0 t 16t 2 feet. Begin by finding the
time at which dsdt 0. Neglect air resistance.]
36. Writing to Learn Suppose you are looking at a graph of
velocity as a function of time. How can you estimate the
acceleration at a given point in time? By estimating the slope of
the velocity graph at that point.
Section 3.4 Velocity and Other Rates of Change 139
37. (b) Speeds up: [1.153, 2.167] and [3,180, ∞] slows down:
[0, 1.153] and [2.167, 3.180]
37. Particle Motion The position (x-coordinate) of a particle moving
on the line y 2 is given by xt 2t 3 13t 2 22t 5 where t
is time in seconds.
(a) Describe the motion of the particle for t 0. See page 140.
(b) When does the particle speed up? slow down?
(c) When does the particle change direction? At t 1.153 sec
and t 3.180 sec
(d) When is the particle at rest? At t 1.153 sec and t 3.180
sec “instantaneously”
(e) Describe the velocity and speed of the particle. See page 140.
(f) When is the particle at the point 5, 2? At about 0.745 sec,
1.626 sec, 4.129 sec
38. Falling Objects The multiflash photograph in Figure 3.34
shows two balls falling from rest. The vertical rulers are marked
in centimeters. Use the equation s 490t 2 (the free-fall
equation for s in centimeters and t in seconds) to answer the
following questions.
(a) How long did it take the balls to fall the first 160 cm? What
was their average velocity for the period? 4/7 of a second. Average
velocity 280 cm/sec
(b) How fast were the balls falling when they reached the 160-
cm mark? What was their acceleration then? Velocity
560 cm/sec; acceleration 980 cm/sec 2
(c) About how fast was the light flashing (flashes per second)?
About 28 flashes per second
39. Writing to Learn Explain how the Sum and Difference Rule
(Rule 4 in Section 3.3) can be used to derive a formula for
marginal profit in terms of marginal revenue and marginal cost.
See page 140.
Standardized Test Questions
You may use a graphing calculator to solve the following
problems.
40. True or False The speed of a particle at t a is given by the
value of the velocity at t a. Justify your answer. False.
It is the absolute value of the velocity.
41. True or False The acceleration of a particle is the second
derivative of the position function. Justify your answer.
42. Multiple Choice Find the instantaneous rate of change of
f (x) x 2 2x 4 at x 1. C
(A) 7 (B) 4 (C) 0 (D) 4 (E) 7
43. Multiple Choice Find the instantaneous rate of change of the
volume of a cube with respect to a side length x. D
(A) x (B) 3x (C) 6x (D) 3x 2 (E) x 3
In Exercises 44 and 45, a particle moves along a line so that its
position at any time t 0 is given by s(t) 2 7t t 2 .
44. Multiple Choice At which of the following times is the
particle moving to the left? E
(A) t 0 (B) t 1 (C) t 2 (D) t 72 (E) t 4
45. Multiple Choice When is the particle at rest? C
(A) t 1 (B) t 2 (C) t 72 (D) t 4 (E) t 5
Explorations
46. Bacterium Population When a bactericide was added to a
nutrient broth in which bacteria were growing, the bacterium
population continued to grow for a while but then stopped
growing and began to decline. The size of the population at
time t (hours) was bt 10 6 10 4 t 10 3 t 2 . Find the
growth rates at t 0, t 5, and t 10 hours. See page 140.
41. True. The acceleration is the first derivative of the velocity which, in turn,
is the second derivative of the position function.