Consider the following case of exponential growth.
The average price of a home in a town was ​$174 comma 000 in 2013 but home prices are rising by 7​% per year.
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Part 1
a. Find an exponential function of the form Upper Q equals Upper Q 0 times left parenthesis 1 plus r right parenthesis Superscript t ​(where rgreater than​0) for growth to model the situation described.
Qequals​$
  
174000times​(1plus
  
0.07​)Superscript t
​(Type an integer or a​ decimal.)
Part 2
b. Fill the table showing the value of the average price of a home for the following five years.
Yearequalst
Average price 
0
​$174 comma 000
1
​$
  
enter your response here
2
​$
  
enter your response here
3
​$
  
enter your response here
4
​$
  
enter your response here
5
​$
  
enter your response here
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

### Part 1: Exponential Function

We are given that:
- The average price of a home in 2013 was $174,000.
- The price is increasing by 7% per year.

The formula for exponential growth is:

\[
Q = Q_0 \times (1 + r)^t
\]

where:
- $ Q_0 $ is the initial quantity (the average price of a home in 2013),
- $ r $ is the rate of growth (7% or 0.07),
- $ t $ is the time in years,
- $ Q $ is the value after time $ t $.

Substituting the given values:

\[
Q = 174,000 \times (1 + 0.07)^t
\]
\[
Q = 174,000 \times (1.07)^t
\]

This is the exponential function that models the situation.

### Part 2: Calculating the Average Price for the Next Five Years

We will now fill out the table using the formula we derived.

For $ t = 0 $, the average price is already given as $174,000.

#### Year 1 ($ t = 1 $):
\[
Q = 174,000 \times (1.07)^1 = 174,000 \times 1.07 = 186,180
\]

#### Year 2 ($ t = 2 $):
\[
Q = 174,000 \times (1.07)^2 = 174,000 \times 1.1449 \approx 199,029
\]

#### Year 3 ($ t = 3 $):
\[
Q = 174,000 \times (1.07)^3 = 174,000 \times 1.225 \approx 213,150
\]

#### Year 4 ($ t = 4 $):
\[
Q = 174,000 \times (1.07)^4 = 174,000 \times 1.3108 \approx 228,527
\]

#### Year 5 ($ t = 5 $):
\[
Q = 174,000 \times (1.07)^5 = 174,000 \times 1.4026 \approx 243,859
\]

### Filled Table:

| Year ($ t $) | Average Price ($ Q $) |
|----------------|-------------------------|
| 0              | $174,000                |
| 1              | $186,180                |
| 2              | $199,029                |
| 3              | $213,150                |
| 4              | $228,527                |
| 5              | $243,859                |

---

Let me know if you need further clarification or more details on any part! Here are some related questions to expand on the topic:

1. What happens to the home price if the growth rate changes to 5% per year?
2. How would you calculate the future price of a home using logarithms?
3. What would the price be after 10 years at a 7% growth rate?
4. How would you model a situation with a decreasing price (negative growth)?
5. Can you explain the difference between exponential growth and linear growth?

**Tip:** When solving exponential growth problems, always ensure that you correctly identify the initial value and the growth rate as this forms the foundation for your formula.

Consider the following case of exponential growth. The average price of a home in a town was $174,000 in 2013 but home prices are rising by 7% per year. Find an exponential function of the form Q = Q0 * (1 + r)^t to model the situation and fill the table showing the value of the average price of a home for the following five years.

Learn how to model the exponential growth of home prices in a town with an initial price of $174,000 in 2013 and a 7% annual increase. This problem includes step-by-step solutions for calculating home prices for the next five years, including the relevant exponential growth formula.

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