Problem

Graph $L(x)~=~20x~+~50$ and $E(x)~=~60(1.1)^x$ for $x~\ge~0$. Find every intersection and interpret each equality for billing periods and cost in dollars.

Coordinate graph showing the linear and exponential billing plans over the feasible nonnegative time window.

Big Picture

What this problem is really about

An intersection means both models produce the same cost at the same billing period, so set their outputs equal and search the entire nonnegative domain. Because the number of crossings is not known in advance, the search must cover the full feasible window. For every crossing found, report an input-output pair and attach periods and dollars to its contextual interpretation.

Answer choices

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Preview only

Use y=20x+50 alone. Its intercept (0 periods, $50) means both plans initially cost $50.

The points (0.71 periods, $64.20) and (22.06 periods, $491.19) show that both plans change at the same rate at those times.

Graph y=20x+50 and y=60(1.1)ˣ for x≥0. They intersect at about (0.71 periods, $64.20) and (22.06 periods, $491.19), so the plans have the same modeled cost at each listed time.

Graph y=20x+50 and y=60(1.1)ˣ for x≥0. They intersect only at about (0.71 periods, $64.20), so the plans have equal cost only then.

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Graph \(L(x)~=~20x~+~50\) and \(E(x)~=~60(1.1)^x\) for \(x~\ge~0\). Find every intersection and interpret each equality for billing periods and cost in dollars.

What this problem is really about

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