A lifeguard at the edge of a circular pool sees a baby fall into the water at another point on the edge of the pool. To save the baby, should he swim directly to her, swim to some point and then walk, or walk all the way around to her? (He has to walk, and not run, because the edge of the pool is slippery.)

The radius of the pool is

feet and the angle to where the baby falls is

,

. The lifeguard can swim

ft/sec and walk

ft/sec, where

.

If he swims at angle

,

, he swims

feet and walks

feet. The total time to reach the baby is thus

seconds.

It takes

seconds to walk all the way and

seconds to swim all the way.

A routine calculus student will compute

and conclude the lifeguard should swim at angle

. But that would put the baby at

*greatest* risk!

A good calculus student knows the minimum time could occur at an endpoint,

or

, and will compare the values at 0,

, and

before making her decision. But she will also compute

and notice it is negative when

. This means the graph of

is concave down, and the minimum must occur at an endpoint. Moreover,

is a

*maximum* when

(assuming

). The

*worst* thing to do is to swim in direction

!

The lifeguard should walk all the way if

or

.

The lifeguard should swim all the way if

or

.

The lifeguard could swim all the way or walk all the way if

or

.

This problem is a slight generalization of a problem in

*A Rigorous Treatment of Maximum-Minimum Problems in the Calculus*, a pamphlet by J. L. Walsh. There,

yds,

(the baby is at the opposite side of the pool),

yds/min and

yds/min. What should the lifeguard do in this case?

What is the effect of

?

Suppose the decision for given

,

, and

, with

, is to swim. Is there always another value of

for which the decision is to walk? Is there always a value of

for which the decision is to walk? (Yes, Yes)

Suppose the decision for given

,

, and

, with

, is to walk. Is there always another value of

,

, for which the decision is to swim? Is there always another value of

for which the decision is to swim? (Yes, No)