Question

The total profit Upper P (x )​(in thousands of​ dollars) from the sale of x hundred thousand pillows is approximated by Upper P (x )equals negative x cubed plus 9 x squared plus 120 x minus 200​, x greater than or equals 5. Find the number of hundred thousands of pillows that must be sold to maximize profit. Find the maximum profit.

Answer 1

Answer:

10 hundred thousands of pillows must be sold to maximize the profit of 900 thousands dollars.

Step-by-step explanation:

We are given the following in the question:

$P(x) = -x^3 + 9x^2 + 120x - 200\\x\geq 5$

Where P(x) is the profit in thousand of dollars and x is the hundred thousand pillows sold.

First, we differentiate P(x) with respect to x, to get,

$\dfrac{d(P(x))}{dx} = \dfrac{d(-x^3 + 9x^2 + 120x - 200)}{dx} = -3x^2 + 18x+120$

Equating the first derivative to zero, we get,

$\dfrac{d(P(x))}{dx} = 0\\\\-3x^2 + 18x+120 = 0$

Solving, with the help of quadratic formula, we get,

$x =-4, x = 10$

Again differentiation P(x), with respect to x, we get,

$\dfrac{d^2(P(x))}{dx^2} = -6x + 18$

At x = 10

$\dfrac{d^2(P(x))}{dx^2} < 0$

Thus,by double differentiation test maxima occurs at x = 10 for P(x).

Thus, 10 hundred thousands of pillows that must be sold to maximize profit.

Maximum Profit

$P(10) = -(10)^3 + 9(10)^2 + 120(10) - 200 = 900$

Thus, the maximum profit is 900 thousands of​ dollars.