A peach grower finds that if he plants 50 trees per acre, each tree will yield 8

Question

A peach grower finds that if he plants 50 trees per acre, each tree will yield 80 bushels. He also estimates that for each additional tree that he plants per acre, the yield of each tree will decrease 2 bushels. How many trees should he plant per acre to maximize his harvest? Hint; the answer may be a negative number A peach grower finds that if he plants 50 trees per acre, each tree will yield 80 bushels. He also estimates that for each additional tree that he plants per acre, the yield of each tree will decrease 2 bushels. How many trees should he plant per acre to maximize his harvest? Hint; the answer may be a negative number

Explanation / Answer

Ans) Let n= the number of additional trees. Let Y= the total yield = number of trees × the yield per tree.
Then:
Y (n) = (50trees + n · trees)( 80 bushels ? n · 2bushels) = (50 + n)(80 ? 2n) = 4000 - 20n - 2n^2
to maximize ! Lets find the critical numbers:
Y'(n) = -20 ? 4n = 0

n = -5 is the only critical number.

Moreover, Y''(n) = ?4

=>R''(-5) = ?4 < 0.

By the second derivative test, Y has a local maximum at n = -5, which is an absolute maximum since it is the only critical number.
The grower should plant 50-5 = 45 trees to maximize the total yield.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.