Respuesta :
Answer:
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.
Step-by-step explanation:
If the no. of trees per acre is doubled then it will be equal to,
[tex]100 \times 2[/tex]
= 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,
[tex]50 \times 200[/tex]
= 10,000 apples per acre
Now , if x trees are planted, the total no. of apples produced per acre will be,
[tex](150 - x) \times (100 + x)[/tex]
= [tex]15000 + 50x - x^{2}[/tex]
= f(x) [say]
So. [tex]\dfrac {df(x)}{dx}[/tex] = f'(x)
= 50 - 2x---(1)
taking f'(x) = 0 , we get,
50 - 2x = 0
⇒ x = 25 --------------(2)
Again,
[tex]\dfrac {d^{2}f(x)}{dx^{2}}[/tex] 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.
