Question

A colony of ants initially has 250,000 ants. The ant population declines at a rate of 2.5% each week.

Write the exponential equation to model the insect population, y, after x weeks.

Answer 1

To model the decline of the ant population, we can use the exponential decay formula:

\[
y = y_0 \cdot (1 - r)^x
\]

Where:
- \(y\) is the population after \(x\) weeks,
- \(y_0\) is the initial population,
- \(r\) is the rate of decline (as a decimal),
- \(x\) is the number of weeks.

Given:
- The initial population \(y_0 = 250,000\),
- The rate of decline \(r = 2.5\% = 0.025\).

Substituting these values into the formula, we have:

\[
y = 250,000 \cdot (1 - 0.025)^x
\]

This simplifies to:

\[
y = 250,000 \cdot (0.975)^x
\]

This is the exponential equation that models the ant population after \(x\) weeks.