Question

natures clean is a company that sells handmade bars of soap. the companys total profit, in dollars, can be modeled by the expression $-\frac{1}{500}(s - 1,100)(s - 3,200)$, where $s$ is the number of bars of soap the company sells.
what do the numbers 1,100 and 3,200 represent?

the profit and revenue from selling 500 bars of soap

the minimum and maximum profit

how many bars of soap the company can sell and exactly break even

how many bars of soap the company can sell to minimize profit and maximize profit

Explanation:

Step1: Understand Profit Function

The profit function is given by \( P(s) = -\frac{1}{500}(s - 1100)(s - 3200) \). In a profit function, the break - even points occur when the profit \( P(s)=0 \).

Step2: Solve for Break - Even Points

Set \( P(s) = 0 \). Then \( -\frac{1}{500}(s - 1100)(s - 3200)=0 \). Since \( -\frac{1}{500}
eq0 \), we set each factor equal to zero:

• If \( s - 1100 = 0 \), then \( s = 1100 \).
• If \( s - 3200=0 \), then \( s = 3200 \).

At the break - even points, the company's profit is zero (total revenue equals total cost). So the values \( s = 1100 \) and \( s = 3200 \) represent the number of bars of soap the company can sell and exactly break even.

Let's analyze the other options:

• Option 1: The expression is about profit as a function of the number of bars sold, not related to selling 500 bars in this context.
• Option 2: The minimum and maximum profit occur at the vertex of the parabola (since the coefficient of \( s^{2}\) is negative, the parabola opens downwards). The vertex occurs at \( s=\frac{1100 + 3200}{2}=\frac{4300}{2}=2150 \), not at 1100 and 3200.
• Option 4: As we saw, 1100 and 3200 are break - even points, not the number of bars to minimize or maximize profit (the vertex gives the number of bars for maximum profit here).

Answer:

how many bars of soap the company can sell and exactly break even