Problem

Profit is $P(x)=x(40-x)-100~\text{dollars}$, where $x(40-x)$ is revenue and $100$ is cost. Interpret a zero of $P$ by writing the output condition, deriving the component equality, and explaining the resulting dollar and break-even meanings.

Big Picture

What this problem is really about

A zero belongs to the combined profit output, not to the revenue component by itself. We’ll set profit equal to zero, substitute the difference model, rearrange it to compare revenue with cost, and use that relationship together with the dollar units to interpret the zero in context.

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A zero means x(40 − x) = 0, so revenue is $0. A profit of $0 can instead occur when revenue equals the $100 cost.

A zero means x = 0, so profit is automatically $0. A function zero sets the output P(x), not the input x, to 0.

A zero has P(x) = 0, so x(40 − x) = 100. Profit is $0, meaning revenue equals cost at a break-even input.

A zero is the y-intercept, so use x = 0 and report −$100. A y-intercept and a function zero are different features.

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Profit is \(P(x)=x(40-x)-100~\text{dollars}\), where \(x(40-x)\) is revenue and \(100\) is cost. Interpret a zero of \(P\) by writing the output condition, deriving the component equality, and explaining the resulting dollar and break-even meanings.

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