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

Question

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

(1 point)
The equation is $$
.

Answer 1

To model the ant population decline using an exponential equation, you can start with the initial population and apply the decay factor based on the percentage decline.

Given:

• Initial population (P₀) = 250,000 ants
• Decline rate = 2.5% = 0.025

The population after \( x \) weeks can be modeled by the following exponential decay equation:

\[ y = P₀(1 - r)^x \]

Where:

• \( y \) is the population after \( x \) weeks.
• \( P₀ \) is the initial population (250,000).
• \( r \) is the decay rate (0.025).

Substituting the values we have:

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

This simplifies to:

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

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

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

Answer 2

not correct

Answer 3

I apologize for any confusion in my previous response. Let's clearly derive the exponential decay model for the ant population.

Given:

• Initial population, \( P_0 = 250,000 \)
• Decline rate = \( 2.5% = 0.025 \)

Exponential Decay Formula:

The general formula for exponential decay is given by:

\[ y = P_0 e^{-kt} \]

Or, using a base that represents the decay per time period:

\[ y = P_0 (1 - r)^t \]

Choosing the Appropriate Model:

In our case, since the population declines by 2.5% each week, the decay rate \( r \) is 0.025. The equation would thus be:

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

Simplifying that:

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

Final Form:

Thus, the correct exponential equation to model the ant population \( y \) after \( x \) weeks is:

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

If you meant a different kind of form or a specific aspect that I should adjust, please let me know!

Answer 4

that number is too long

Answer 5

I apologize for any misunderstanding. Let's simplify the notation to focus on the essential parts of the exponential decay equation without additional context.

The exponential equation modeling the ant population \( y \) after \( x \) weeks, given a decline rate of 2.5% per week, is:

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

This equation captures the initial population and the rate of decline in a straightforward format. Please let me know if you need further assistance!