The total profit P(x) (in thousands of dollars) from a sale of x thousand units

Question

The total profit P(x) (in thousands of dollars) from a sale of x thousand units of a new product u given by P(x) = ln (-x^3 + 9x^2 + 48x + 1) (0 lessthanorequalto x lessthanorequalto 10). a) Find the number of units that should be sold in order to maximize the total profit. b) What is the maximum profit? a) The number of units that should be sold in order to maximize the total profit is F. (Simplify your answer.) b) The maximum profit is approximately $. (Do not round until the final answer. Then round to the nearest dollar as needed.)

Explanation / Answer

(a) P(x) = ln(-x^3 + 9x^2 + 48x + 1)

P'(x) = (-3x^2 + 18x + 48)/(-x^3 + 9x^2 + 48x + 1)

P'(x0 = 0

-3x^2 + 18x + 48 = 0

-x^2 + 6x + 16 = 0

(x - 8)(x + 2) = 0

x = 8

Number of units = 8000

(b) Maximum profit = ln(-8^3 + 9 * 8^2 + 48 * 8 + 1) = 6.107 thousand = $6107

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