Question

(3 points) There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees ?

Answer 1

15 trees should be added to the existing orchard in order to maximize the total output of trees.

Explanation:

Number of trees = 50

Number of apples each tree gives = 800

For each additional tree planted in the orchard, the output per tree drops by 10 apples. Let x be the number of trees added, then,

Number of apple is given by:

`V(x) = (50+x)(800-10x) = -10x^2+300x + 40000`

First, we differentiate V(x) with respect to x, to get,

`\displaystyle(d(V(x)))/(dx) = (d(-10x^2+300x + 40000))/(dx) = -20x + 300`

Equating the first derivative to zero, we get,

`\displaystyle(d(V(x)))/(dx) = 0\\\\-20x + 300 = 0`

Solving, we get,

`-20x + 300 = 0\\\\x=\displaystyle(300)/(20) = 15`

Again differentiation V(x), with respect to x, we get,

`\displaystyle(d^2(V(x)))/(dx^2) = -20`

At x = 15

`\displaystyle(d^2(V(x)))/(dx^2) < 0`

Thus, by double derivative test, the maxima occurs at x = 15 for V(x).

Thus, 15 trees should be added to the existing orchard in order to maximize the total output of trees.

Maximum output of apples =

`V(15) = (50+15)(800-10(15)) = 42250`