A farmer finds that if she plants 75 trees per acre, each tree will yleld 60 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?
trees
Give your answer rounded to the nearest whole number.

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UpStudy AI · Accepted Answer

Let $$ x $$ be the number of additional trees planted beyond the base rate of 75 trees per acre. Then:

- The total number of trees per acre is 
  $$
  T = 75 + x.
  $$
- The yield per tree decreases by 3 bushels for each additional tree, so the yield per tree is 
  $$
  Y_{\text{per tree}} = 60 - 3x.
  $$
- The total harvest (in bushels) per acre is 
  $$
  H = (75 + x)(60 - 3x).
  $$

Because the decrease is stated only for each additional tree planted, the situation is modeled for $$ x \ge 0 $$. We now express the harvest as a function of $$ x $$:
$$
H(x) = (75 + x)(60 - 3x).
$$

Evaluate the harvest at $$ x = 0 $$ (which corresponds to 75 trees per acre):
$$
H(0) = 75 \times 60 = 4500.
$$

Now, consider what happens if we plant one additional tree (i.e. $$ x=1 $$):
$$
H(1) = (75+1)(60-3(1)) = 76 \times 57 = 4332.
$$

Since $$ H(1) < H(0) $$ and the relationship applies only when planting additional trees (i.e. $$ x \ge 0 $$), the maximum total harvest occurs at $$ x = 0 $$.

Thus, the farmer should plant 
$$
75 \text{ trees per acre}
$$
to maximize her harvest.

Rounded to the nearest whole number, the answer is $$ 75 $$ trees per acre.
The farmer should plant 75 trees per acre to maximize her harvest.

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