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.

(1 point)
The equation is $$y=·.

Answer 1

To model the ant population \( y \) after \( x \) weeks, given that the population declines at a rate of 2.5% each week, we can use the formula for exponential decay:

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

where:
- \( y_0 \) is the initial population,
- \( r \) is the decay rate (in decimal form),
- \( x \) is the number of weeks,
- \( y \) is the population after \( x \) weeks.

Given:
- \( y_0 = 250,000 \)
- \( r = 0.025 \) (which is 2.5% expressed as a decimal)

Therefore, the equation becomes:

\[
y = 250000 \cdot (1 - 0.025)^x
\]

Simplifying \( (1 - 0.025) \):

\[
y = 250000 \cdot (0.975)^x
\]

So the exponential equation to model the insect population after \( x \) weeks is:

\[
y = 250000 \cdot (0.975)^x
\]