Answer to Question #270755 in Algebra for stella

Question

1.     A farmer finds that if she plants 75 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 3 bushels. How many trees should she plant per acre to maximize her harvest? Also, find a rigorous algebraic solution for the problem.

Answer 1

Consider a farmer plants 75 trees per acre, each tree will yield 20 bushels of fruit.

Let x be the number of trees, than for each new tree, 3 fewer bushels are yielded.

So for (75 + x) trees, there are (20-3x) bushels.

The total amount of fruit in bushels is,

$F(x) = (75+x)(20-3x) \\ = -3x^2 -205x +1500$

Since the coefficient of x² is negative, the parabola opens downward and has a maximum value. To find the price that maximizes the fruit price, begin by finding the x-value of the vertex,

$h = \frac{-b}{2a} \\ = -\frac{-205}{2(-3)} \\ =-\frac{205}{6}$

The number of trees should she plant per acre to maximize her harvest is,

$75 +x = 75 -\frac{205}{6} \\ = \frac{245}{6} \\ = 41$

Thus, the number of trees per acre to maximize the harvest is 41.

F(75) = 25

F(78) = 28

F(81) = 31

slope = 1

y = x+50