A farmer finds that if she plants 45 trees per acre, each
tree will yield 65 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.

Question

UpStudy AI · Accepted Answer

Let $$ x $$ be the number of trees planted per acre. We are told that if $$ x=45 $$ then each tree yields 65 bushels. In addition, for each additional tree (that is, for each tree beyond 45) the yield of each tree decreases by 3 bushels. (The problem does not state that the yield increases by the same amount if fewer than 45 trees are planted, so the given relationship is intended only for additional trees.)

When $$ x \ge 45 $$, the yield per tree is given by
$$
\text{Yield per tree} = 65 - 3\,(x-45).
$$
Thus, the total harvest $$ P(x) $$ (in bushels per acre) is
$$
P(x) = x\Bigl[65-3\,(x-45)\Bigr].
$$

Now simplify the expression inside the brackets:
$$
65 - 3\,(x-45) = 65 - 3x + 135 = 200-3x.
$$
So the production function becomes
$$
P(x)= x(200-3x)=200x-3x^2.
$$

If we were allowed to plant any number of trees, including fewer than 45, we would maximize $$ P(x)=200x-3x^2 $$ by computing the vertex of this quadratic. Its vertex is at
$$
x=\frac{-200}{2(-3)}=\frac{200}{6}\approx33.33.
$$
This value is less than 45, and indeed plugging $$ x\approx33.33 $$ would give a larger total harvest mathematically. However, the problem statement specifies that the decrease in yield applies to “each additional tree planted”—that is, the yield adjustment is given only for increasing the number of trees beyond 45. There is no information indicating that if the farmer plants fewer than 45 trees the yield per tree would increase above 65 bushels.

Thus, since the farmer’s estimate applies only when adding trees (i.e. $$ x\ge45 $$), she should not plant any trees in excess of 45.

Therefore, to maximize her harvest under the conditions given, the farmer should plant

$$
\boxed{45} 
$$
trees per acre.
The farmer should plant 45 trees per acre to maximize her harvest.

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