maximum profit calc

Question

The profit from an Almond orchard is given by  dollars, where  is the number of almond trees per acre. How many trees per acre will maximize the profit?

a) The number of trees should be

b) The maximum profit would be

Answer 1

To find the maximum, take the derivative and set it equal to zero.

P(x) = x(630 - 3x) = 630x - 3x². The derivative is P'(x) = 630 - 6x. Set it equal to zero and solve:

630 - 6x = 0

6x = 630

x = 105

So, the maximum profit is attained when x = 105 almond trees per acre. That maximum profit is determined by plugging in x = 105 into the original equation: P(105) = (105)(630 - 3(105)) = $33,075