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Question 26
A farmer finds that if she plants 65 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre,
the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest?
Give your answer rounded to the nearest whole number.
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UpStudy AI · Accepted Answer

To maximize the harvest, we need to find the optimal number of trees to plant per acre. Let's denote the number of trees planted per acre as $$ x $$.

Given:
- If 65 trees are planted per acre, each tree yields 20 bushels of fruit.
- For each additional tree planted per acre, the yield of each tree decreases by 2 bushels.

Let's set up the equation for the total yield of fruit per acre:
$$ \text{Total Yield} = 20 \times 65 $$

Since the yield decreases by 2 bushels for each additional tree, the yield per tree can be represented as:
$$ \text{Yield per Tree} = 20 - 2(x - 65) $$

The total yield per acre can be represented as:
$$ \text{Total Yield} = 20 \times 65 - 2(x - 65)^2 $$

To maximize the total yield, we need to find the value of $$ x $$ that maximizes the total yield. This can be done by finding the critical points of the total yield function and determining which one corresponds to the maximum.

Let's calculate the total yield function and find the critical points.
Simplify the expression by following steps:
- step0: Solution:
  $$20\times 65-2\left(x-65\right)^{2}$$
- step1: Multiply the numbers:
  $$1300-2\left(x-65\right)^{2}$$
- step2: Expand the expression:
  $$1300-2x^{2}+260x-8450$$
- step3: Subtract the numbers:
  $$-7150-2x^{2}+260x$$
It seems there was an issue with calculating the derivative. Let's calculate it manually.

The derivative of the total yield function with respect to $$ x $$ is:
$$ \frac{d}{dx}(-7150 - 2x^2 + 260x) = -4x + 260 $$

To find the critical points, we set the derivative equal to zero and solve for $$ x $$:
$$ -4x + 260 = 0 $$
$$ -4x = -260 $$
$$ x = \frac{-260}{-4} $$
$$ x = 65 $$

Therefore, the critical point is $$ x = 65 $$. This means that the maximum yield occurs when 65 trees are planted per acre.

Since we are asked to round the answer to the nearest whole number, the farmer should plant 65 trees per acre to maximize her harvest.
The farmer should plant 65 trees per acre to maximize her harvest.

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