(b) Find the maximum profit.
(c) Find the price(s) that would enable the company to break even. If there is more than one price, use the "and" button
(b) To find the maximum profit, we need to determine the price that will generate the highest profit. From the profit equation we derived earlier:
P(x) = (15 - x)(2000 - 40x)
To maximize profit, we need to find the critical points of the profit function. Taking the derivative of P(x) with respect to x and setting it equal to zero:
P'(x) = -40x + 2000 - 40x - 15 = 0
-80x + 2000 = 15
-80x = -1985
x = 24.8125
This critical point lies within the feasible domain of x, so we need to determine the profit at x = 24.8125:
P(24.8125) = (15 - 24.8125)(2000 - 40(24.8125)) = (-9.8125)(1075) = -10551.5625
Since this profit value is negative, it indicates that the price of $24.8125 will not yield a profit. Therefore, the company should not sell the product at this price.
(c) To find the price(s) that would enable the company to break even, we need to set the profit function equal to zero and solve for x:
(15 - x)(2000 - 40x) = 0
15 - x = 0 or 2000 - 40x = 0
x = 15 or x = 50
Therefore, the company will break even if they sell the product at a price of $15 or $50.