Question

Over the past ten years, the town's population doubled in size. The population is currently 12,000. Use the variable, P, to represent the town’s original population. Set up an equation and find the original population of the town ten years ago.

In the form of a paragraph, explain in complete sentences the steps necessary to set up the equation. Include the final equation in your explanation. Complete your work in the space provided or upload a file that can display math symbols if your work requires it.

Answer 1

Let's call the initial population \(C\). If the population \(P\) doubles every year \(x\), that means when \(x=1\) (1 year after year 0, the initial year), you have a population of \(P=2C\). The next year, when \(x=2\), you have \(P=2(2C)=2^2C\). When \(x=3\), you have \(P=2(2^2C)=2^3C\).

Generalizing from this pattern, you'll notice that the population at year \(x\) is given by
\[P=2^xC\]
Notice that when \(x=0\), you have the initial population: \(P=2^0C=C\).

This means the equation that models the population \(P\) over time \(x\) (in years) is given by
\[P=C(2^x)\]

Answer 2

You're given that the current population (after 10 years, which means when \(x=10\)) is 12,000. Solve for \(C\):
\[12,000=C(2^{10})\]