Let's determine which given function best represents the value of the real estate after [tex]\( t \)[/tex] years using the information provided in the problem:
### Step 1: Determine the Appreciation Rate
First, we need to find the annual appreciation rate. Given:
- Value after 1 year: \[tex]$200,038.80
- Value after 2 years: \$[/tex]208,040.35
We calculate the appreciation rate by dividing the value at the end of the second year by the value at the end of the first year:
[tex]\[
\text{Appreciation Rate} = \frac{\text{Value after 2 years}}{\text{Value after 1 year}} \approx \frac{208040.35}{200038.80} \approx 1.04
\][/tex]
So, the annual appreciation rate is approximately 1.04, meaning a 4% increase per year.
### Step 2: Establish the General Form of the Function
Assuming the property appreciates at a constant rate, the value of the property after [tex]\( t \)[/tex] years can be expressed as:
[tex]\[
\text{Value} = \text{Initial Value} \times (\text{Appreciation Rate})^t
\][/tex]
Given the initial purchase price is \[tex]$192,345 and our appreciation rate is approximately 1.04, we substitute these values into the formula:
\[
f(t) = 192345 \times (1.04)^t
\]
### Step 3: Evaluating the Given Options
Now, let's evaluate the options provided with \( t = 2 \) to check which one correctly represents the value after 2 years:
1. \( f(t) = 200,038.80 \times (1.04)^t \)
- For \( t = 2 \):
\[
f(2) = 200038.80 \times (1.04)^2 \approx 216361.97
\]
2. \( f(t) = 200,038.80 \times (0.04)^t \)
- For \( t = 2 \):
\[
f(2) = 200038.80 \times (0.04)^t \approx 320.06
\]
3. \( f(t) = 192,345 \times (0.04)^2 \)
- This evaluates to a constant number:
\[
f(2) = 192345 \times (0.04)^2 = 192345 \times 0.0016 \approx 307.75
\]
4. \( f(t) = 192,345 \times (1.04)^t \)
- For \( t = 2 \):
\[
f(2) = 192345 \times (1.04)^2 \approx 208040.35
\]
### Step 4: Conclusion
Comparing these values to the actual value given for the property after 2 years, which is \$[/tex]208,040.35, we see that the fourth option most accurately represents the value after [tex]\( t \)[/tex] years.
Thus, the function that best represents the value of the real estate after [tex]\( t \)[/tex] years is:
[tex]\[
f(t) = 192,345 \times (1.04)^t
\][/tex]
