A peach orchard owner wants to maximize the amount of peaches produced by his or

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A peach orchard owner wants to maximize the amount of peaches produced by his orchard. He has found that the per-tree yield is equal to 1000 whenever he plants 40 or fewer trees per acre, and that when more than 40 trees are planted per acre, the per- tree yield decreases by 30 peaches per tree for every extra tree planted. For example, if there were 35 trees planted per acre, each tree would produce 1000 peaches. If there were 45 trees planted per acre, each tree would produce 1000 - 30 * (45 - 40) peaches, which is roughly equal to 850 peaches. Find the function that describes the per-tree yield, Y, in terms of x. Y = ? if x is no more than 40 trees per acre y=? if x is greater than 40 trees per acre Find the total yield per acre, T, that results from planting x trees per acre. T = ? if a- is no more than 40 trees per acre T = ? if x is greater than 40 trees per acre Differentiate T with respect to x dT/ dx = ? if x is less than 40 trees per acre dT/ dx = ? ifx fa greater than 40 trees per acre Does this derivative ever equal zero? Yes/No Sketch the graph of T as x varies and hence find the value of x that maximizes the yield and the maximum value of the yield. Optimal value of x : ? trees per acre Maximum yield : ? peaches per acre Is T differentiable when x equals 40? YesNo

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