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.
