Calculus: Analytic Geometry and Calculus, with Vectors

5.2 Trends, maxima, and minima 303

Prove that L is maximum when an appropriate angle 0 satisfies the equation
cot 0 = 0.
33 A spherical ball of radius r settles slowly into a full conical glass of water
and causes an overflow of water. The glass has height a, and the lines on the
surface of the glass make the angle 0 with the axis of the glass. Find the radius
r for which the overflow is a maximum. Remark: This problem is famous because
it is difficult enough to be remembered and discussed by those who have solved
it. Solution: With or without careful scrutiny of other cases, wesuppose that
the ball is neither so small that it can be completely submerged nor so large that
it will fail to be tangent to the glass when it is in its lowest position. Letting
0, C, and B be the vertex of the conical glass, the center of the ball, and the
bottom of the ball, we see that 1OCl = r csc 0 and 15-BI = r(csc 0- 1). The
submerged segment of the ball has thickness h, where

(1) h = a - r(csc 0 - 1).

The overflow (measured in cubic units) is equal to the volume Y of this segment,

and Problem 2 of Problems 4.59 shows that

(2) Y =rh2(3r - h) = Tr[h2r -h21.

Differentiation gives

(3) h2+2hrdhA

A]

-h2 -

= 7rh[ 2r dh + h (1

• dr J J

Using (1) and the formula for dh/dr calculated from it gives

(4)

dY it

dr sin2 0[a sin 0 - (sin 0 + cos 20)r].

Since a sin 0 and (sin 0 + cos 20) are positive when 0 < 0 < 7r/2, it follows that

P is a maximum when

(5) a sin 0

sin 0 + cos 20

34 When distances are measured in feet, the equation of the path followed

by water projected from our fire hose is

(1 + m2)x2

y=mx- 100

where m is the slope of the path at the nozzle which is located at the origin. Find

the value of m for which the water will reach the greatest height on a wall 40 feet

from the nozzle and find the greatest height. Partial ans.: One of the two answers

is 9.

35 Remark: The following big-government problem need not be taken too

seriously; its purpose is to neutralize a problem involving a country that allowed

its unemployed boomerang repairmen to starve to death. Determine the num-

ber of officials that must be supported in a country containing n workers, and