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;
[tex]F'(x) = \dfrac{d}{dx}( -3x^2- 130x + 1000)[/tex]
[tex]F'(x) = ( -3(2x)- 130(1) +0)[/tex]
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
[tex]F''(x) = \dfrac{d}{dx}( -6x- 130)[/tex]
[tex]F''(x) = ( -6(1))[/tex]
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
[tex]\simeq[/tex] 28
Therefore, the number of trees per acre to maximize the harvest is 28
