Question

A home gardener estimates that 24 apple trees will have an average yield of 104 apples per tree. But because of the size of the garden, for each additional tree planted the yield will decrease by two apples per tree. (a) How many additional trees should be planted to maximize the total yield of apples

Answer 1

The farmer should plant 14 additional trees, for maximum yield.

Explanation:

Given

`Trees = 24`

`Yield = 104`

`x \to additional\ trees`

So, we have:

`Trees = 24 + x`

`Yield = 104 - 2x`

Required

The additional trees to be planted for maximum yield

The function is:

`f(x) = Trees * Yield`

`f(x) = (24 + x) * (104 - 2x)`

Open bracket

`f(x) = 24 * 104 + 104x - 24 * 2x - x * 2x`

`f(x) = 2796 + 104x - 48x - 2x^2`

`f(x) = 2796 + 56x - 2x^2`

Rewrite as:

`f(x) = - 2x^2 + 56x + 2796`

Differentiate

`f'(x) = -4x + 56`

Equate
`f'(x) = -4x + 56` to 0 and solve for x to get the maximum of x

`-4x + 56 = 0`

`-4x =- 56`

Divide by -4

`x =14`

The farmer should plant 14 additional trees, for maximum yield.