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Answer :
Sure! Let's take a look at solving the problem step by step.
The problem involves calculating the future value of the house in Texas after "t" years, given that the initial value of the house is [tex]$156,000 and it appreciates by 12% annually.
When we talk about appreciation or compound growth, we typically use the formula:
\[ \text{Future Value} = \text{Initial Value} \times (1 + \text{Rate})^t \]
where:
- \( \text{Initial Value} \) is the current value of the asset, in this case, $[/tex]156,000.
- [tex]\( \text{Rate} \)[/tex] is the appreciation rate, which is 12% or 0.12 as a decimal.
- [tex]\( t \)[/tex] is the number of years.
Now, let's evaluate each expression from the options to see if they match this formula.
### Option (A): [tex]\( 156,000 \cdot (0.12)^r \)[/tex]
- This expression is incorrect because it uses [tex]\( 0.12 \)[/tex] instead of [tex]\( 1.12 \)[/tex] to represent the appreciation factor. Additionally, it uses [tex]\( r \)[/tex] instead of [tex]\( t \)[/tex].
### Option (B): [tex]\( 156,000 \cdot (1.12)^T \)[/tex]
- This expression is correct. It uses the appreciation factor [tex]\( 1.12 \)[/tex] appropriately and raises it to the power of [tex]\( T \)[/tex], which represents the number of years.
### Option (C): [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
- This expression is correct. It explicitly adds [tex]\( 1 + 0.12 \)[/tex] to get the appreciation factor [tex]\( 1.12 \)[/tex], and raises it to the power of [tex]\( t \)[/tex].
### Option (D): [tex]\( 156,000 \cdot (1-0.12)^1 \)[/tex]
- This expression is incorrect. It uses subtraction instead of addition, representing a depreciation, not appreciation.
### Option (E): [tex]\( 156,000 \cdot \left(1+\frac{0.12}{12}\right)^4 \)[/tex]
- This expression is incorrect for this context. It appears to be set up for monthly compounding over 4 months, but the problem specifies annual appreciation.
So, the expressions that could be used to calculate the value of the house after [tex]\( t \)[/tex] years, based on annual appreciation, are:
- Option (B): [tex]\( 156,000 \cdot (1.12)^T \)[/tex]
- Option (C): [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
I hope this helps! Let me know if you have any more questions.
The problem involves calculating the future value of the house in Texas after "t" years, given that the initial value of the house is [tex]$156,000 and it appreciates by 12% annually.
When we talk about appreciation or compound growth, we typically use the formula:
\[ \text{Future Value} = \text{Initial Value} \times (1 + \text{Rate})^t \]
where:
- \( \text{Initial Value} \) is the current value of the asset, in this case, $[/tex]156,000.
- [tex]\( \text{Rate} \)[/tex] is the appreciation rate, which is 12% or 0.12 as a decimal.
- [tex]\( t \)[/tex] is the number of years.
Now, let's evaluate each expression from the options to see if they match this formula.
### Option (A): [tex]\( 156,000 \cdot (0.12)^r \)[/tex]
- This expression is incorrect because it uses [tex]\( 0.12 \)[/tex] instead of [tex]\( 1.12 \)[/tex] to represent the appreciation factor. Additionally, it uses [tex]\( r \)[/tex] instead of [tex]\( t \)[/tex].
### Option (B): [tex]\( 156,000 \cdot (1.12)^T \)[/tex]
- This expression is correct. It uses the appreciation factor [tex]\( 1.12 \)[/tex] appropriately and raises it to the power of [tex]\( T \)[/tex], which represents the number of years.
### Option (C): [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
- This expression is correct. It explicitly adds [tex]\( 1 + 0.12 \)[/tex] to get the appreciation factor [tex]\( 1.12 \)[/tex], and raises it to the power of [tex]\( t \)[/tex].
### Option (D): [tex]\( 156,000 \cdot (1-0.12)^1 \)[/tex]
- This expression is incorrect. It uses subtraction instead of addition, representing a depreciation, not appreciation.
### Option (E): [tex]\( 156,000 \cdot \left(1+\frac{0.12}{12}\right)^4 \)[/tex]
- This expression is incorrect for this context. It appears to be set up for monthly compounding over 4 months, but the problem specifies annual appreciation.
So, the expressions that could be used to calculate the value of the house after [tex]\( t \)[/tex] years, based on annual appreciation, are:
- Option (B): [tex]\( 156,000 \cdot (1.12)^T \)[/tex]
- Option (C): [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
I hope this helps! Let me know if you have any more questions.
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