There is a garden of apples and the profit from a garden of apple trees is given by
f( x ) = 70 + 30 x^2 - (1/3) x^3,
where x = number of apples trees per acre. How many trees per acre will maximize the profit?
f'(x) = 60x-x^2 = x(60-x)
so, max profit when x=60
Solution:
max{70+30 x^2-x^3/3} = 36070
is at x = 60
To find the number of apple trees per acre that will maximize the profit, we need to find the critical points of the profit function. The critical points occur where the derivative of the function is equal to zero.
First, let's find the derivative of the profit function f(x) with respect to x:
f'(x) = 2(30)x - 3(1/3)x^2
Simplifying:
f'(x) = 60x - x^2
Next, set the derivative equal to zero and solve for x:
60x - x^2 = 0
Rearranging the equation:
x^2 - 60x = 0
Factoring out an x:
x(x - 60) = 0
From this equation, we can see that either x = 0 or x - 60 = 0.
Since we are looking for the number of trees per acre, x cannot be 0 (since we need at least one tree). Therefore, we only consider the second equation:
x - 60 = 0
Solving for x:
x = 60
So, the number of trees per acre that will maximize the profit is 60 trees.