Question

An apple orchard has an average yield of 36 bushels of apples per tree if tree density is 26 trees per acre. For each unit increase (decrease) in tree density, the yield decreases (increases) by 2 bushels per tree. How many trees per acre should be planted to maximize the yield?

Answer 1

Answer:

4 trees per acre

Step-by-step explanation:

Given,

The original number of apples per tree = 36 bushels,

Original density of a tree = 26 per acre,

Since, for each decrease in tree density, the yield increases by 2 bushels per tree.

Let x be the number of units decreased in tree density,

So, new density = 26 - x,

New number of apples = 36 + 2x

Thus, the total yield would be,

P(x) = Number of apples per tree × density of a tree

⇒ P(x) = (26 - x)(36 + 2x)

⇒ P(x) = 936 + 52x - 36x - 2x²,

⇒ P(x) = 936 + 16x - 2x²

Differentiating with respect to x,

P'(x) = 16 - 4x

Again differentiating with respect to x,

P''(x) = -4

For maxima or minima,

P'(x) = 0

⇒ 16 - 4x = 0

⇒ -4x = -16

⇒ x = 4

For x = 4, P''(x) = negative,

Hence, 4 trees per acre should be planted to maximize the yield.