Consider the following case of exponential decay. Complete parts​ (a) through​ (c) below.
A privately owned forest that had 4 comma 000 comma 000 acres of old growth is being clear cut at a rate of 3​% per year.
Question content area bottom
Part 1
a. Create an exponential function of the form Upper Q equals Upper Q 0 times left parenthesis 1 plus r right parenthesis Superscript t​, ​(where rgreater than0 for growth and rless than0 for​ decay) to model the situation described. 
Qequals
  
4000000times​(
  
0.97​)Superscript t
​(Type integers or​ decimals.)
Part 2
b. Create a table showing the value of the quantity Q for the first 10 years of growth.
Yearequalst
Acres 
Yearequalst
Acres
0
4 comma 000 comma 000
6
  
enter your response here
1
  
enter your response here
7
  
enter your response here
2
  
enter your response here
8
  
enter your response here
3
  
enter your response here
9
  
enter your response here
4
  
enter your response here
10
  
enter your response here
5
  
enter your response here
​(Round to the nearest whole number as​ needed.)

To address the exponential decay scenario you described, we'll break it down into parts (a) and (b).

### Part 1: Exponential Function
The formula for exponential decay is given as:

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

Here:
- \( Q_0 = 4,000,000 \) (the initial number of acres)
- \( r = -0.03 \) (the rate of decay, since it's decreasing by 3% per year)

Thus, the function becomes:

\[
Q = 4,000,000 \times (0.97)^t
\]

### Part 2: Table of Values
Now, we will compute the number of acres remaining for the first 10 years.

\[
Q(t) = 4,000,000 \times (0.97)^t
\]

We'll calculate this for \( t = 0 \) to \( t = 10 \):

- **Year 0**: 
  \[
  Q(0) = 4,000,000 \times (0.97)^0 = 4,000,000
  \]

- **Year 1**: 
  \[
  Q(1) = 4,000,000 \times (0.97)^1 \approx 3,880,000
  \]

- **Year 2**: 
  \[
  Q(2) = 4,000,000 \times (0.97)^2 \approx 3,773,600
  \]

- **Year 3**: 
  \[
  Q(3) = 4,000,000 \times (0.97)^3 \approx 3,669,088
  \]

- **Year 4**: 
  \[
  Q(4) = 4,000,000 \times (0.97)^4 \approx 3,566,244
  \]

- **Year 5**: 
  \[
  Q(5) = 4,000,000 \times (0.97)^5 \approx 3,465,052
  \]

- **Year 6**: 
  \[
  Q(6) = 4,000,000 \times (0.97)^6 \approx 3,365,442
  \]

- **Year 7**: 
  \[
  Q(7) = 4,000,000 \times (0.97)^7 \approx 3,267,370
  \]

- **Year 8**: 
  \[
  Q(8) = 4,000,000 \times (0.97)^8 \approx 3,170,752
  \]

- **Year 9**: 
  \[
  Q(9) = 4,000,000 \times (0.97)^9 \approx 3,075,558
  \]

- **Year 10**: 
  \[
  Q(10) = 4,000,000 \times (0.97)^{10} \approx 2,981,744
  \]

### Summary Table

| Year (t) | Acres      |
|----------|------------|
| 0        | 4,000,000  |
| 1        | 3,880,000  |
| 2        | 3,773,600  |
| 3        | 3,669,088  |
| 4        | 3,566,244  |
| 5        | 3,465,052  |
| 6        | 3,365,442  |
| 7        | 3,267,370  |
| 8        | 3,170,752  |
| 9        | 3,075,558  |
| 10       | 2,981,744  |

Would you like any further details or explanations? Here are some related questions to consider:

1. How does changing the decay rate affect the function?
2. What would happen if the forest were to regrow at a certain rate?
3. Can you graph the exponential decay function?
4. How long would it take for the forest to drop below 2,000,000 acres?
5. What real-world factors could influence the rate of decay?

**Tip:** Always round your final answers to the nearest whole number when dealing with populations or physical quantities like acres.

Consider the following case of exponential decay. A privately owned forest that had 4,000,000 acres of old growth is being clear-cut at a rate of 3% per year.

Part 1
a. Create an exponential function of the form Q = Q0 * (1 + r)^t to model the situation.

Part 2
b. Create a table showing the value of the quantity Q for the first 10 years of growth.

Learn how to model exponential decay with a real-world example involving a forest with 4,000,000 acres, clear-cut at 3% per year. Find the exponential function representing the remaining forest area over time and create a 10-year decay table. Suitable for high school algebra students.

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