James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

applied project Where To Sit at the Movies 465
where v 0 − vss 0d and v 1 − vss 1d are the velocities of the object at the positions s 0
and s 1. Hint: By the Chain Rule,
m dv
dt − m dv
ds
ds
dt
− mv
dv
ds
(b) How many foot-pounds of work does it take to throw a baseball at a speed of
90 miyh?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly
to the catcher with an initial velocity of 100 ftys. Assume that the velocity vstd of
the ball after t seconds satisfies the differential equation dvydt − 2 1
10 v because of air
resistance. How long does it take for the ball to reach home plate? (Ignore any vertical
motion of the ball.)
(b) The manager of the team wonders whether the ball will reach home plate sooner if it
is relayed by an infielder. The shortstop can position himself directly between the outfielder
and home plate, catch the ball thrown by the outfielder, turn, and throw the ball
to the catcher with an initial velocity of 105 ftys. The manager clocks the relay time
of the shortstop (catching, turning, throwing) at half a second. How far from home
plate should the shortstop position himself to minimize the total time for the ball to
reach home plate? Should the manager encourage a direct throw or a relayed throw?
What if the shortstop can throw at 115 ftys?
; (c) For what throwing velocity of the shortstop does a relayed throw take the same time
as a direct throw?
APPLIED Project
CAS
where to sit at the movies
25 ft
10 ft
9 ft
å
x
¨
4 ft
A movie theater has a screen that is positioned 10 ft off the floor and is 25 ft high. The first
row of seats is placed 9 ft from the screen and the rows are set 3 ft apart. The floor of the
seating area is inclined at an angle of − 20° above the horizontal and the distance up the
incline that you sit is x. The theater has 21 rows of seats, so 0 < x < 60. Suppose you decide
that the best place to sit is in the row where the angle subtended by the screen at your eyes
is a maximum. Let’s also suppose that your eyes are 4 ft above the floor, as shown in the
figure. (In Exercise 4.7.78 we looked at a simpler version of this problem, where the floor is
horizontal, but this project involves a more complicated situation and requires technology.)
1. Show that
− arccosS a 2 1 b 2 2 625
2ab
where a 2 − s9 1 x cos d 2 1 s31 2 x sin d 2
and b 2 − s9 1 x cos d 2 1 sx sin 2 6d 2
D
2. Use a graph of as a function of x to estimate the value of x that maximizes . In which
row should you sit? What is the viewing angle in this row?
3. Use your computer algebra system to differentiate and find a numerical value for the
root of the equation dydx − 0. Does this value confirm your result in Problem 2?
4. Use the graph of to estimate the average value of on the interval 0 < x < 60. Then
use your CAS to compute the average value. Compare with the maximum and minimum
values of .
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