Question

3. In a single growing season at the Smith Family Orchard, the average yield per apple

tree is 150 apples when the number of trees per acre is 100. For each additional tree
over 100, the average yield per tree decreases by 1.
a. What would be the average yield per tree if the number of trees per acre was
doubled? What would be the total yield in that case?
b. How many trees should be planted per acre to maximize the total yield?

Answer 1

a. Average yield per tree will be 50 apples. Total yield in that case will be 10,000 apples per acre.

b. 25 more trees are to be planted per acre to maximize the total yield.

Explanation:

If the no. of trees per acre is doubled then it will be equal to,

`100 * 2`

= 200

So, the no. of trees per acre is increased by (200 - 100) = 100

hence, the average yield per tree will be,

(150 - 100) = 50 in that case.

Hence, the total yield will be,

`50 * 200`

= 10,000 apples per acre

Now , if x trees are planted, the total no. of apples produced per acre will be,

`(150 - x) * (100 + x)`

=
`15000 + 50x - x^(2)`

= f(x) [say]

So.
`\frac {df(x)}{dx}` = f'(x)

= 50 - 2x---(1)

taking f'(x) = 0 , we get,

50 - 2x = 0

⇒ x = 25 --------------(2)

Again,

`\frac {d^(2)f(x)}{dx^(2)}` at x = 25

= -2 (<0)

So, f(x) attains it's maximum value at x = 25

Hence,

25 more trees are to be planted per acre to maximize the total yield.