waterworks is a company that manufactures and sells paddleboards. its profit p, in hundreds of dollars…

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)= - 2x^3+32x^2 - 56x. 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 and fewer than paddleboards.

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)= - 2x^3+32x^2 - 56x. 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 and fewer than paddleboards.

Answer

Explanation:

Step1: Set the profit function equal to zero.

We want to find the zeros of $P(x)= - 2x^{3}+32x^{2}-56x$. Set $P(x) = 0$, so $-2x^{3}+32x^{2}-56x=0$.

Step2: Factor out the greatest - common factor.

Factor out $-2x$ from the left - hand side: $-2x(x^{2}-16x + 28)=0$.

Step3: Factor the quadratic expression.

Factor $x^{2}-16x + 28$. We need two numbers that multiply to $28$ and add up to $-16$. The numbers are $-2$ and $-14$. So $x^{2}-16x + 28=(x - 2)(x - 14)$. Then our equation becomes $-2x(x - 2)(x - 14)=0$.

Step4: Use the zero - product property.

Set each factor equal to zero:

  • $-2x=0$ gives $x = 0$.
  • $x - 2=0$ gives $x = 2$.
  • $x - 14=0$ gives $x = 14$.

The zeros of the function are $x = 0$, $x = 2$, and $x = 14$. Since $x$ is the number of paddleboards in thousands and the company must produce more than $0$ to make a profit, and the profit function is negative for $x>14$ (we can test a value like $x = 15$: $P(15)=-2\times15^{3}+32\times15^{2}-56\times15=-2\times3375+32\times225-840=-6750 + 7200-840=-390$), the company must produce more than $2000$ and fewer than $14000$ paddleboards to make a profit.

Answer:

More than $2$ (in thousands) and fewer than $14$ (in thousands)