Question

waterworks is a company that manufactures and sells paddleboards. its profit p, in hundreds of dollars earned, is a function of the number of paddleboards sold x, measured in thousands. profit is modeled by the function ( p(x) = -4x^3 + 48x^2 - 80x ). what do the zeros of the function tell you about the number of paddleboards that waterworks should produce? in order to make a profit, waterworks must produce more than (square) and fewer than (square) paddleboards.

Explanation:

Step1: Factor out common term

$P(x) = -4x(x^2 - 12x + 20)$

Step2: Factor quadratic expression

$P(x) = -4x(x - 2)(x - 10)$

Step3: Find zeros of the function

Set $P(x)=0$:
$-4x(x - 2)(x - 10)=0$
Solutions: $x=0$, $x=2$, $x=10$

Step4: Analyze profit-positive interval

For a cubic function with negative leading coefficient, $P(x)>0$ when $010$. But $x>10$ leads to decreasing profit (negative), so valid interval is $2

Answer:

In order to make a profit, Waterworks must produce more than $\boldsymbol{2000}$ and fewer than $\boldsymbol{10000}$ paddleboards.