Wampum If an apple orchard occupies one acre of land, how many trees should it contain so as to product the largest apple crop? This is an example of an optimization problem. The word optimum means "best possible" especially, the best under a given set of conditions. This project seeks an optimum by analyzing a simple mathematical model of the orchard. The mode] is the function that describes how the total yield depends on the number of trees. An immediate inipise is to just plant a lot of trees, using the idea that more trees 2 more apples. However, there is a catch: if there are too many trees in a single acre, they crowd together. Each tree then gets less sunlight and nutrients, so it produces fewer apples. For example. the relation between yield per tree, YIN), and the 'l'tifi?' of trees, N , may be like the following graph pounds Y per tree 750 1. In words, describe what the graph in the gure above tells us about apple production per tree. 2. Our goal is to choose N so that the total yield, T{N), will be as large as possible. Give an expression relating T{N), YIN), and N. 3. Since the graph of Y(N} given in Figure 1 isn't precisely known, let's replace the Y{N) graph in Figure 1 by the following approximation shown in Figure 2. answer the following questions using the graph given in Figure 2. (a) Find a formula for the straight segment of the new graph of Y(N] on the interval 40 E N g: 180. (b) What is the formula for T(N) on the same interval? (c) What are the formulas for T(N) when 0 5 N 5 40 and when N 2 180? Graph T as a function of N and describe the graph in words. {(1) Fill in the following blanloa: l' , N540 T(N): , 4U1sn