300 Functions, graphs, and numbers
15 Show that of all rectangles having a given area, a square has the least
perimeter.
16 Find the radius and height of the cone of greatest volume that can be
made from a circular disk of radius a by cutting out (or folding over, as chemists
do) a sector and bringing the edges of the remaining part together.
17 An ordinary tomato can is to be constructed to have a given volume Y.
Determine the height h and radius r of the can for which total
surface area is a minimum.
18 As in Figure 5.294, the base and lateral surface of a
solid right circular cone are tangent to a sphere of radius a.
Find the height of the solid having minimum volume.
Outline of solution: The height y and base radius r are related
by the formula
a _ ry - a r2+y2
Figure 5.294
which equates two expressions for sin B. Squaring, solving
for r2, and using the formula Y = irr2y for the volume of the solid, we find thatira2 y2 dY ira2y y - 4a
Y 3 y - 2a' dy_
3 (y - 2a)2The conditions of the problem require that y > 2a. If 2a < y < 4a, then
dY/dy < 0 and V is decreasing. If y > 4a, then dY/dy > 0 and V is increasing.
Thus Y is minimum when y = 4a.
19 Supposing that xi, x2, , x, are givennumbers, find the values of x,
if any, for which
n
I (x - xk) 2
k=1attains maximum and minimum values.
20 The elementary theory of probability tells us that the number pn,k defined
bypn.knl
= k!(n- k)!pk(l - p)n-kis the probability of exactly k successes in it trials when p is the probability of
success in each trial. Supposing that n and k are given integers for which n > 0
and 0 < k 5 n, find the number p which maximizes pn.k Hint: Ignore the
numerical coefficient and find the p which maximizes pk(1 - p)'-k. Ans.:
p = k/n.
21 An observant senator observes that if he hires just one secretary, she will
work nearly 30 hours per week but that each additional secretary produces con-
versations that reduce her effectiveness. In fact, if there are x secretaries, x not
exceeding 30, then each one will work only30 - 30