Questions: A farmer finds that if she plants 65 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest? trees Give your answer

A farmer finds that if she plants 65 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest?
trees
Give your answer

Transcript text: A farmer finds that if she plants 65 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest? $\square$ trees Give your answer

Solution

Solution Steps

Step 1: Formulate the total yield function

The total yield of the harvest, represented as a function of $x$, is given by $Y = x(y_0 - d(x - x_0))$. This simplifies to $Y = xy_0 - dx^2 + dx_0x$.

Step 2: Simplify the function and find the derivative

The simplified yield function is $Y = xy_0 - dx^2 + dx_0x$. Taking the derivative with respect to $x$ gives $Y' = y_0 - 2dx + dx_0$.

Step 3: Solve for $x$ by setting the derivative to zero

Setting $Y'$ to zero gives $0 = y_0 - 2dx + dx_0$, which simplifies to $x = \frac{y_0 + dx_0}{2d}$.