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CHAPTER 4 • •
Applications of Differentiation
278 P2: OSO/OVY QC: OSO/OVY T1: OSO October 25, 2018 17:24 4-68
R
A
+
V
1
θ 1 -
2 - x x
x θ 2 30. In an AC circuit with voltage V(t) = v sin(2 ft), a voltmeter
actually shows the average (root-mean-square) voltage of
1 √
v∕ 2. If the frequency is f = 60 (Hz) and the meter reg-
isters 115 volts, find the maximum voltage reached.
B
31. A Norman window has the outline of a semicircle on top of
Exercise 25 a rectangle. Suppose there is 8 + mile of wood trim avail-
able. Discuss why a window designer might want to maxi-
26. Suppose that light reflects off a mirror to get from point A mize the area of the window. Find the dimensions of the
to point B as indicated in the figure. Assuming a constant rectangle (and, hence, the semicircle) that will maximize
velocity of light, we can minimize time by minimizing the the area of the window.
distance traveled. Find the point on the mirror that min-
imizes the distance traveled. Show that the angles in the
figure are equal (the angle of incidence equals the angle of
reflection).

A

B
2
1 32. Suppose a wire 2 ft long is to be cut into two pieces, each
θ 1 θ 2
of which will be formed into a square. Find the size of each
piece to maximize the total area of the two squares.
x 4 - x 33. An advertisement consists of a rectangular printed region
Exercise 26
plus 1-in margins on the sides and 2-in margins at top and
2
27. The human cough is intended to increase the flow of air to bottom. If the area of the printed region is to be 92 in , find
the lungs, by dislodging any particles blocking the wind- the dimensions of the printed region and overall advertise-
pipe and changing the radius of the pipe. Suppose a wind- ment that minimize the total area.
pipe under no pressure has radius r . The velocity of air 34. An advertisement consists of a rectangular printed region
0
through the windpipe at radius r is approximately V(r) = plus 1-in margins on the sides and 1.5-in margins at top
2
cr (r − r) for some constant c. Find the radius that maxi- and bottom. If the total area of the advertisement is to be
0
mizes the velocity of air through the windpipe. Does this 120 in , what dimensions should the advertisement be to
2
mean the windpipe expands or contracts? maximize the area of the printed region?
28. To supply blood to all parts of the body, the human artery 35. A hallway of width a = 5 ft meets a hallway of width b =
system must branch repeatedly. Suppose an artery of radius 4 ft at a right angle. (a) Find the length of the longest ladder
r branches off from an artery of radius R (R > r) at an angle that could be carried around the corner. (Hint: Express the
. The energy lost due to friction is approximately length of the ladder as a function of the angle in the figure.)

E( ) = csc + 1 − cot .
r 4 R 4 θ b
Find the value of that minimizes the energy loss.
a
29. In an electronic device, individual circuits may serve many
purposes. In some cases, the flow of electricity must be con- (b) Show that the maximum ladder length for general a and
trolled by reducing the power instead of amplifying it. The b equals (a 2∕3 + b 2∕3 3∕2 . (c) Suppose that a = 5 and the lad-
)
power absorbed by the circuit is der is 8 ft long. Find the minimum value of b such that the
ladder can turn the corner. (d) Solve part (c) for a general a
2
p(x) = V x , and ladder length L.
(R + x) 2
36. A company’s revenue for selling x (thousand) items is given Copyright © McGraw-Hill Education
for a voltage V and resistance R. Find the value of x that by R(x) = 35x − x 2
maximizes the power absorbed. x + 35 . (a) Find the value of x that maximizes
2

298 | Lesson 4-7 | Optimization