Swim, Swim and Walk, or Walk?
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.)[more]
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)[less]
Former Harvard Professor J. L. Walsh emphasized rigor in solving max-min problems. In this case, the principles are:
A function continuous on a closed bounded interval has a (global) minimum there.
A function differentiable at cannot have a minimum there unless .
Consequently, a function continuous on has a miminum value, and the minimum can occur only at an endpoint or where or where does not exist.
J. L. Walsh, A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, Boston: Heath, 1962.