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vazorg [7]
3 years ago
13

A farmer finds that if she plants 50 trees per acre, each tree will yield 80 bushels of fruit. She estimates that for each addit

ional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?
Mathematics
1 answer:
pashok25 [27]3 years ago
6 0

Answer:

28

Step-by-step explanation:

From the given information:

Let x be the number of trees.

F(x) = (50 +x) (20 - 3x)

F(x) = 1000 - 150x + 20x - 3x²)

F(x) = -3x² - 130x + 1000

Differentiating F(x) with respect to x;

F'(x) = \dfrac{d}{dx}( -3x^2- 130x + 1000)

F'(x) = ( -3(2x)- 130(1) +0)

F'(x) = -6x -130

Now; we set F'(x) to be equal to zero to determine the critical value;

-6x - 130 = 0

x = - 130/6

Differentiating F''(x) with respect to x

F''(x) = \dfrac{d}{dx}( -6x- 130)

F''(x) = ( -6(1))

F''(x) = -6 (<0)

Thus; by the second derivative, the revenue function F(x) is maximum when x = -130/6

Therefore, the number of trees she should plant per acre to maximize her harvest is:

50 + x = 50 - 130/6

= 85/3

\simeq 28

Therefore, the number of trees per acre to maximize the harvest is 28

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