Question

profit function
a business’s profit is modeled by the function ( p(x) = -3(x - 2)(x - 14) ), where ( x ) represents the number of items produced in hundreds and ( p(x) ) represents the profit in thousands of dollars.
the company begins to be profitable if it sells (\boldsymbol{\text{dropdown}}) items.
the maximum profit is (\boldsymbol{\text{dropdown}}) dollars.

Explanation:

Step1: Find break-even point (profit=0)

Set $P(x)=0$, so $-3(x-2)(x-14)=0$. Solve for $x$: $x=2$ or $x=14$. Since profit becomes positive after $x=2$, and $x$ is hundreds of items, calculate actual items: $2\times100=200$.

Step2: Identify maximum profit value

From the graph, the vertex is $(8,108)$. $P(x)$ is profit in thousands of dollars, so calculate actual profit: $108\times1000=108000$.

Answer:

The company begins to be profitable if it sells 200 items.
The maximum profit is 108000 dollars.