Q:

A farmer finds that if she plants 100 trees per acre, each tree will yield 35 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Accepted Solution

A:
Answer:126 trees should she plant per acre to maximize her harvest.Step-by-step explanation:Let x be the additional tress that must be plant after planting 100 tress.So, total number of trees = 100 + x treesAlso, Each additional tree planted decreases the yield of each tree by 3 bushels. So , net bushels by each tree = 35 - 3xThe revenue function becomesf(x) = (100 + x)(35 - 3x)Thus, Differentiating it by using chain rule as:f'(x) = (100 + x)35 + 100(35 - 3x)f'(x) = 3500 + 35x + 3500 - 300xf'(x) = 7000 Β - 265xFor maxima of f(x), f'(x) = 07000 Β - 265x = 0Thus, x = 26.4151Since, x represents tress, so, x is 26.So, total tress she would plant to earn maximum revenue = 100 + 26 = 126 trees