https://file.aiquickdraw.com/mathbot/file/1729566036695v7weqlqv.jpg

Let's go through the questions in the image step by step, focusing on how each relates to exponential growth or decay.

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### Part (a):
**A town's population increases by 1.9% per year.**

**i. What is the population's 10-year growth factor?**

To calculate the growth factor over a number of years, we use the formula:
\[
\text{Growth Factor} = (1 + r)^t
\]
where:
- \(r\) is the growth rate per year (in decimal form),
- \(t\) is the time in years.

Here, \(r = 0.019\) (1.9% increase per year), and \(t = 10\) years.

So, the 10-year growth factor is:
\[
(1 + 0.019)^{10} = (1.019)^{10}
\]

**ii. What is the population's 10-year percent change?**

To find the percent change, we can use:
\[
\text{Percent Change} = (\text{Growth Factor} - 1) \times 100
\]
where the Growth Factor is the value calculated in part i.

---

### Part (b):
**The value of your car changes by -13% per year.**

**i. What is your car value's 3-year decay factor?**

For decay, we use a similar formula, but with a negative rate. The decay factor is:
\[
\text{Decay Factor} = (1 - r)^t
\]
where:
- \(r = 0.13\) (13% annual decrease),
- \(t = 3\) years.

So, the 3-year decay factor is:
\[
(1 - 0.13)^{3} = (0.87)^3
\]

**ii. What is your car value's 3-year percent change?**

Again, the percent change can be calculated as:
\[
\text{Percent Change} = (\text{Decay Factor} - 1) \times 100
\]

Since this is a decrease, the percent change will be negative.

---

### Part (c):
**The mass of a bacteria colony changes by 23% per day.**

**i. What is the 9-day growth factor for the bacteria colony's mass?**

We use the same growth factor formula:
\[
\text{Growth Factor} = (1 + r)^t
\]
where:
- \(r = 0.23\) (23% daily increase),
- \(t = 9\) days.

So, the 9-day growth factor is:
\[
(1 + 0.23)^{9} = (1.23)^9
\]

**ii. What is the 9-day percent change for the bacteria colony's mass?**

Finally, the percent change is calculated as:
\[
\text{Percent Change} = (\text{Growth Factor} - 1) \times 100
\]

---

Would you like me to compute the exact numerical values for each part?

Here are 5 related questions:
1. How would the formulas change if the population decreased by a percentage annually?
2. What happens to the decay factor if the rate of decay increases?
3. How can we interpret the growth factor if the population growth is compounded more frequently than annually (e.g., monthly)?
4. What is the effect of doubling the time period on the growth or decay factor?
5. How would exponential decay apply to the value of an investment that loses 5% per year?

**Tip**: Exponential models are very common in real life, especially in finance and population studies. Recognizing growth and decay processes can help in predicting future outcomes!

Complete the following assuming that each situation can be modeled by an exponential function. a. Suppose a town's population increases by 1.9% per year. i. What is the population's 10-year growth factor? ii. What is the population's 10-year percent change? b. Suppose the value of your car changes by -13% per year. i. What is your car value's 3-year decay factor? ii. What is your car value's 3-year percent change? c. The mass of a bacteria colony in an experiment changes by 23% per day. i. What is the 9-day growth factor for the bacteria colony's mass? ii. What is the 9-day percent change for the bacteria colony's mass?

Solve exponential growth and decay problems involving population growth, car value depreciation, and bacterial colony mass changes. This problem covers key exponential models and provides step-by-step solutions for different growth and decay rates.

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