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4-69 QC: OSO/OVY T1: OSO November 6, 2018 15:33 SECTION 4.7 • • Optimization 279 Unsaved...
the revenue and find the maximum revenue. (b) For any
positive constant c, find x to maximize R(x) = cx − x 2 . EXPLORATORY EXERCISES
x + c
2
1. In a preliminary investigation of Kepler’s cask problem (sec-
3
2
37. In t hours, a worker makes Q(t) =−t + 12t + 60t items. tion 4.3), you showed that a height-to-diameter ratio (x∕y)
′
Graph Q (t) and explain why it can be interpreted as the effi- of √ 2 for a cylindrical barrel will maximize the volume
ciency of the worker. (a) Find the time at which the worker’s (see Figure a). However, real casks are bowed out. Kepler
efficiency is maximum. (b) Let T be the length of the work- approximated a cask with the straight-sided barrel in Fig-
day. Suppose that the graph of Q(t) has a single inflection ure b. It can be shown (we told you Kepler was good!)
point for 0 ≤ t ≤ T, called the point of diminishing returns. that the volume of this barrel is V = [y + (w − y) +
2
2
2
Show that the worker’s efficiency is maximized at the point √ 3
2
2
of diminishing returns. y(w − y)] z − w . Treating w and z as constants, show that
′
V (y) = 0 if y = w∕2. Recall that such a critical point can
38. Suppose that group tickets to a concert are priced at AED correspond to a maximum or minimum of V(y) or some-
40 per ticket if 20 tickets are ordered, but cost AED 1 per thing else (e.g., an inflection point). To discover what we
ticket less for each extra ticket ordered, up to a maximum of have here, redraw Figure b to scale (show the correct rela-
50 tickets. (For example, if 22 tickets are ordered, the price tionship between 2y and w). In physical terms (think about
is AED 38 per ticket.) (a) Find the number of tickets that increasing and decreasing y), argue that this critical point
maximizes the total cost of the tickets. (b) If management is neither a maximum nor minimum. Interestingly enough,
wanted the solution to part (a) to be 50, how much should such a nonextreme critical point would have a definite ad-
the price be discounted for extra tickets ordered? vantage to the Austrians. Their goal was to convert the mea-
surement z into an estimate of the volume. Explain why
39. In sports where balls are thrown or hit, the ball often fin- ′
ishes at a different height than it starts. Examples include V (y) = 0 means that small variations in y would convert
a downhill golf shot and a basketball shot. In the diagram, to small errors in the volume V.
a ball is released at an angle and finishes at an angle
above the horizontal (for downhill trajectories, would be
negative). Neglecting air resistance and spin, the horizontal z 2y
range is given by
2
2
2v cos 2x
R = (tan − tan )
g
FIGURE a
if the initial velocity is v and g is the gravitational constant.
In the following cases, find to maximize R (treat v and g as
◦
◦
constants): (a) = 10 , (b) = 0 and (c) =−10 . Verify z w 2y
◦
◦
◦
that = 45 + ∕2 maximizes the range.
FIGURE b
θ β 2. A ball is thrown from s = b to s = a (where a < b) with ini-
tial speed v . Assuming that air resistance is proportional
0
to speed, the time it takes the ball to reach s = a is
1 ( b − a )
T =− ln 1 − c ,
c v 0
where c is a constant of proportionality. A baseball player
is 300 ft from home plate and throws a ball directly toward
x 2 y 2 home plate with an initial speed of 125 ft/s. Suppose that
40. The area enclosed by the ellipse a 2 + b 2 = 1 equals ab.
Find the maximum area of a rectangle inscribed in the el- c = 0.1. How long does it take the ball to reach home plate?
Another player standing x feet from home plate has the op-
lipse (that is, a rectangle with sides parallel to the x-axis and tion of catching the ball and then, after a delay of 0.1 s, re-
y-axis and vertices on the ellipse). Show that the ratio of the laying the ball toward home plate with an initial speed of
maximum inscribed area to the area of the ellipse to the area 125 ft/s. Find x to minimize the total time for the ball to
of the circumscribed rectangle is 1 : :2.
2 reach home plate. Is the straight throw or the relay faster?
41. Show that the maximum volume enclosed by a right circular What, if anything, changes if the delay is 0.2 s instead of
1
cylinder inscribed in a sphere equals √ times the volume 0.1 s? For what length delay is it equally fast to have a re-
3
lay and not have a relay? Do you think that you could catch
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education 42. Find the maximum area of an isosceles triangle of given is considered important to have a relay option in baseball?
of the sphere.
and throw a ball in such a short time? Why do you think it
perimeter p. [Hint: Use Heron’s formula for the area of a tri-
Repeat the above if the second player throws the ball with
√
initial speed 100 ft/s. For a delay of 0.1 s, find the value of
angle of sides a, b and c : A =
s(s − a)(s − b)(s − c), where
1
the initial speed of the second player’s throw for which it is
s = (a + b + c).]
2
equally fast to have a relay and not have a relay.
299
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