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We are given an exponential growth model for the population of pavement ants in Taylorsville, UT:

\[
P_n = P_0 (1 + r)^n
\]

Where:
- \(P_0 = 4600\) (initial population),
- \(P_n = 15000\) (population after 10 years),
- \(n = 10\) (time in years),
- \(r\) is the growth rate we need to find.

We can now solve for \(r\).

### Step 1: Solve for \(1 + r\)
Start by substituting the values into the equation:

\[
15000 = 4600 (1 + r)^{10}
\]

Divide both sides by 4600:

\[
\frac{15000}{4600} = (1 + r)^{10}
\]

\[
\approx 3.2609 = (1 + r)^{10}
\]

### Step 2: Take the 10th root to isolate \(1 + r\)

\[
1 + r = \left(3.2609\right)^{1/10}
\]

Using this, we can calculate \(1 + r\).

### Step 3: Subtract 1 to find \(r\) (growth rate)

Now, let’s calculate both \(1 + r\) and \(r\) and express them to the required precision.### Solution:

(a) The growth multiplier \(1 + r\) is approximately **1.125** (rounded to the nearest thousandth).

(b) The growth rate \(r\) as a percentage is approximately **12.5%** (rounded to the nearest tenth).

Do you need further details or have any questions?

Here are some related questions for further understanding:
1. How would the population change if the growth rate doubled?
2. What would the population be after 15 years, assuming the same growth rate?
3. How would the model change if the population started at 5000 instead of 4600?
4. How can we adjust the formula for varying growth rates each year?
5. What is the doubling time for this population given the growth rate?

**Tip:** In exponential models, even small changes in the growth rate can lead to large differences over time!

Suppose a population of pavement ants in Taylorsville, UT has an initial population of 4600 and 10 years later the population reaches 15000. Use an explicit exponential model to find the rate of growth and common ratio for the pavement ant population. (a) Express the growth multiplier as a decimal to the nearest thousandth. (b) Express the growth rate as a percentage to the nearest tenth.

Learn how to calculate the growth rate and multiplier for a population of pavement ants in Taylorsville, UT, growing from 4600 to 15000 in 10 years. Includes step-by-step calculations using exponential growth formulas. Ideal for students in Grades 9-12.

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