In the last section, we found that for is a reasonable choice of a function for the amount of nectar collected by a bee that stays on a flower for time , where is measured in minutes.

We also decided that an *optimal* time is exactly the right amount of time that a bee should spend at a flower.

Recall that the bee wants to collect as much nectar as possible over the course of a day. In order to do so, the bee must maximize the *rate* at which she collects nectar. In other words, the bee must collect the most food in the shortest amount of time.

Recall that the bee spends time not only collecting nectar at a flower, but also flying from flower to flower. Thus, the bee must actually maximize the *average rate per visit*. The average rate per visit includes both the time spent at the flower as well as the time spent getting to the flower in the first place.

We can call this average rate per visit . The units are the amount of nectar collected per unit time. Denote the time spent traveling from flower to flower by , and let be the number of flowers visited. Then,

To better understand the function , let's take and examine the resulting ,

Sketch a graph of for and estimate the value of time where the average rate of food intake is maximized.

ShowUse calculus methods to determine the exact time at which the rate of food intake is maximized.

ShowWe have that . Applying the quotient rule, we have,

Setting the derivative equal to zero gives

which implies that

That is, the function has a local maximum or minimum at

From the graph above, we know that we have found a local maximum instead of a local minimum at . Use the first derivative test to prove that we have indeed found a local maximum.

ShowWe check the sign of for and for .

For , we have

which is less than as the numerator is negative while the denominator is positive.

For , we have

which is greater than .

That is, to the left of , but to the right of . This indicates that we have found a local maximum at which corresponds to with our estimate obtained from the graph above.

Thus, a bee should stay a flower for minutes, given that the travel time to the next flower is minute.

We have answered our question. That is, given that and that , we know that the bee should stay at the flower for minutes. In reality, however, the function may have a different form and may not be exactly minute. In the next section, we shall repeat the steps we have taken here but with an unknown and .