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Answer :
To solve this problem, we need to calculate the value of a condominium after a certain number of years, given that its value appreciates annually. Here's how we can approach this:
1. Determine the Initial Investment:
The initial purchase price of the condominium is $156,000.
2. Understand the Appreciation Rate:
The home value in the neighborhood appreciates by 12% annually. To use this in calculations, we convert this percentage to a decimal by dividing by 100, which gives us an appreciation rate of 0.12.
3. Find the Expression for Appreciated Value:
The general formula for calculating the future value of an appreciating asset is:
[tex]\[
\text{Future Value} = \text{Initial Value} \times (1 + \text{Appreciation Rate})^t
\][/tex]
where [tex]\( t \)[/tex] is the number of years.
4. Apply the Formula:
Plugging in the values, we have:
[tex]\[
\text{Future Value} = 156,000 \times (1 + 0.12)^t = 156,000 \times (1.12)^t
\][/tex]
This expression correctly calculates the future value of the condominium.
5. Analyze the Choices:
Given the options:
- Option A: [tex]\( 156,000 \cdot (0.12)^t \)[/tex] – This is incorrect because it uses the appreciation rate alone, rather than adding it to 1.
- Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex] – This is correct as it matches our derived expression.
- Option C: [tex]\( 156,000 \cdot (1 \neq 0.12)^t \)[/tex] – This makes no sense because of incorrect syntax and the use of logical operators.
- Option D: [tex]\( 156,000 \cdot (1-0.12)^t \)[/tex] – This is incorrect because it would imply depreciation rather than appreciation.
- Option E: [tex]\( 156,000 \cdot\left(1+\frac{0.12}{12}\right)^{\prime} \)[/tex] – This appears to misinterpret the problem by suggesting monthly calculation without a correct format for [tex]\( t \)[/tex].
Therefore, the correct expression that represents the value of the house after [tex]\( t \)[/tex] years is Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex].
1. Determine the Initial Investment:
The initial purchase price of the condominium is $156,000.
2. Understand the Appreciation Rate:
The home value in the neighborhood appreciates by 12% annually. To use this in calculations, we convert this percentage to a decimal by dividing by 100, which gives us an appreciation rate of 0.12.
3. Find the Expression for Appreciated Value:
The general formula for calculating the future value of an appreciating asset is:
[tex]\[
\text{Future Value} = \text{Initial Value} \times (1 + \text{Appreciation Rate})^t
\][/tex]
where [tex]\( t \)[/tex] is the number of years.
4. Apply the Formula:
Plugging in the values, we have:
[tex]\[
\text{Future Value} = 156,000 \times (1 + 0.12)^t = 156,000 \times (1.12)^t
\][/tex]
This expression correctly calculates the future value of the condominium.
5. Analyze the Choices:
Given the options:
- Option A: [tex]\( 156,000 \cdot (0.12)^t \)[/tex] – This is incorrect because it uses the appreciation rate alone, rather than adding it to 1.
- Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex] – This is correct as it matches our derived expression.
- Option C: [tex]\( 156,000 \cdot (1 \neq 0.12)^t \)[/tex] – This makes no sense because of incorrect syntax and the use of logical operators.
- Option D: [tex]\( 156,000 \cdot (1-0.12)^t \)[/tex] – This is incorrect because it would imply depreciation rather than appreciation.
- Option E: [tex]\( 156,000 \cdot\left(1+\frac{0.12}{12}\right)^{\prime} \)[/tex] – This appears to misinterpret the problem by suggesting monthly calculation without a correct format for [tex]\( t \)[/tex].
Therefore, the correct expression that represents the value of the house after [tex]\( t \)[/tex] years is Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex].
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