Problem 10 (Good Samaritan Problem)

A baker, being a good samaritan, bakes cookies for needy children. He figures the number of cookies he can bake per hour is proportional to the product of the number of hours of sleep he gets in a day times times the number of hours he spends baking cookies. Assuming he is either sleeping or baking cookies, how many hours of sleep should the baker get each day in order to maximize the number of cookies he can bake in a day?

Solution
If the number of cookies N the baker can make per hour is proportional to the number t of hours the baker bakes times the number of hours he sleeps (he sleeps 24 - t hours a day), then

N = k t (24 - t)

where k is a constant of proportionality. Hence the total number of cookies the baker makes every day is

T = t N = k t ² (24 - t )

To find the value of t that maximizes T, we find the derivative

dT/dt = k t (48 - 3t)

and find the values of t in the interval [0, 24] where dT/dt = 0. Finding these, we get t = 0, 16. Since the second derivative d2T/dt2 is less than 0 on the interval [0, 24], we can resort to the second derivative test and conclude the point t = 16 is a maximum point. Hence, the baker should work 16 hours a day and sleep 8 hours a day. Of course, you can't determine the actual number of cookies he makes per day since we didn't tell you the contant of proportionality k .

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Last modified on Tuesday, January 12, 1999