There are 50 apple trees in an orchard. Each | vedclass

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(C) Let $x$ be the number of additional trees planted.The total number of trees becomes $(50 + x)$.The production per tree becomes $(800 - 10x)$.The t

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$15$

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(C) Let $x$ be the number of additional trees planted.The total number of trees becomes $(50 + x)$.The production per tree becomes $(800 - 10x)$.The total production $P(x)$ is given by the product of the number of trees and the production per tree:$P(x) = (50 + x)(800 - 10x)$$P(x) = 40000 - 500x + 800x - 10x^2$$P(x) = -10x^2 + 300x + 40000$To maximize the output,we find the derivative $P'(x)$ and set it to $0$:$P'(x) = \frac{d}{dx}(-10x^2 + 300x + 40000) = -20x + 300$Setting $P'(x) = 0$:$-20x + 300 = 0$$20x = 300$$x = 15$To verify this is a maximum,we check the second derivative:$P''(x) = -20$Since $P''(x) < 0$,the function has a maximum at $x = 15$.

There are $50$ apple trees in an orchard. Each tree produces $800$ apples. For each additional tree planted in the orchard,the output per tree drops by $10$ apples. The number of trees that should be added to the existing orchard to maximize the total output is:

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