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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

Respuesta :

15 more  trees should be a planted to maximize the total output of apples in the orchard.

Step-by-step explanation:

Here, the total number of trees planted  = 50 apple trees

The produce of each tree = 800 apples

So, the total apples = Total trees x Produce of each tree  = 50 x 800

                                 = 40,000

Let us assume the additional trees planted  = m

So, now the total number of trees now  = 50 + m

As produce decrease 10 times  per tree  = 10 m

So, the average produce per tree  = 800 - 10 m

So, the new produce P   =  (50+m)(800-10 m)

or, P =  (40000 - 500 m + 800 m -10 m²)

⇒P =  -10 m²  +300 m  + 40000

or P =  -m² + 30 m  + 4000

Maximum occurs when m  = [tex]\frac{-b}{2a} = \frac{-30}{-2} = 15[/tex]

So, the Value of P is maximum when m = 15

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

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