Question

On a citrus farm, the average annual yield of oranges is 124 pounds per tree when the number of trees is 22 per acre. For each additional tree over 22, the annual yield per tree decreases by 2 pounds per tree. How many orange trees should be planted per acre to maximize the number of pounds of oranges per acre

Answer 1

Answer:

Number of orange trees = 3528

Step-by-step explanation:

Given: Average annual orange yield = 124 pounds (lbs) per tree

Number of trees = 22 per acre

Each tree produces 124 pounds, so 22 trees will produce

124*22 = 2728 lbs per acre

For every additional tree over 22, 2 pounds per tree is lost.

So, if you add x trees, you lose 2x pounds.

Thus, we have the function:

f(x) = (124 - 2x)*(22 + x) for the added tree(s)

$f(x) = 2728 + 80x - 2x^{2}$

For quadratic equations like this, the maximum(or minimum) is obtained by substituting x= -b/2a into the given equation. a and b are obtained from the general form of a quadratic equation:

$y =ax^{2} + bx + c$

Comparing this with f(x), we have that

a = -2 (coefficient of x2)

b = 80 (coefficient of x)

Using these values in x = -b/2a, we get

$x = -\frac{80}{2*(-2)} = 20$

So, use x = 20 in the maximum function we got earlier

$f(x) = 2728 + 80(20) -2(20^{2}) = 2728 + 1600 - 800\\ \\f(x) = 3528$

Therefore, the number of oranges that should be planted in order to maximize the number of pounds of oranges per acre = 3528