Answer :
To solve this problem, we need to determine which expressions correctly calculate the value of a condominium that appreciates by 12% annually. Let's analyze each option based on the details given:
1. Expression: [tex]\(156,000 \cdot \left(1+\frac{0.12}{12}\right)^t\)[/tex]
- This expression suggests that the appreciation is compounded monthly. Since the problem states that it's an annual appreciation, compounding monthly would not be the correct approach if we're looking for annual appreciation. This is not a valid expression for annual appreciation.
2. Expression: [tex]\(156,000 \cdot (1-0.12)^t\)[/tex]
- This expression implies a depreciation of 12% annually, which means it decreases the value of the property each year. Since the value is actually increasing, this expression is not applicable.
3. Expression: [tex]\(156,000 \cdot (1.12)^t\)[/tex]
- This expression uses the formula for appreciating an amount annually by 12%, where [tex]\(1.12\)[/tex] indicates a 12% increase (1 + 0.12). This is a correct way to calculate the value of the house after [tex]\(t\)[/tex] years with annual appreciation.
4. Expression: [tex]\(156,000 \cdot (0.12)^t\)[/tex]
- This implies compounding only the growth percentage without adding it to the original value (base 1), which doesn't give the total amount but rather a tiny fraction over time. This doesn't correctly represent annual appreciation.
5. Expression: [tex]\(156,000 \cdot (1+0.12)^t\)[/tex]
- This expression simplifies to [tex]\(156,000 \cdot (1.12)^t\)[/tex], which is the same as expression 3 and correctly describes an annual appreciation of 12%. This is indeed a valid expression for calculating the appreciated value.
Based on this analysis, the expressions that could be used to calculate the value of the house after [tex]\(t\)[/tex] years are:
- [tex]\(156,000 \cdot (1.12)^t\)[/tex]
- [tex]\(156,000 \cdot (1+0.12)^t\)[/tex]
These expressions account for the annual appreciation of 12%.
1. Expression: [tex]\(156,000 \cdot \left(1+\frac{0.12}{12}\right)^t\)[/tex]
- This expression suggests that the appreciation is compounded monthly. Since the problem states that it's an annual appreciation, compounding monthly would not be the correct approach if we're looking for annual appreciation. This is not a valid expression for annual appreciation.
2. Expression: [tex]\(156,000 \cdot (1-0.12)^t\)[/tex]
- This expression implies a depreciation of 12% annually, which means it decreases the value of the property each year. Since the value is actually increasing, this expression is not applicable.
3. Expression: [tex]\(156,000 \cdot (1.12)^t\)[/tex]
- This expression uses the formula for appreciating an amount annually by 12%, where [tex]\(1.12\)[/tex] indicates a 12% increase (1 + 0.12). This is a correct way to calculate the value of the house after [tex]\(t\)[/tex] years with annual appreciation.
4. Expression: [tex]\(156,000 \cdot (0.12)^t\)[/tex]
- This implies compounding only the growth percentage without adding it to the original value (base 1), which doesn't give the total amount but rather a tiny fraction over time. This doesn't correctly represent annual appreciation.
5. Expression: [tex]\(156,000 \cdot (1+0.12)^t\)[/tex]
- This expression simplifies to [tex]\(156,000 \cdot (1.12)^t\)[/tex], which is the same as expression 3 and correctly describes an annual appreciation of 12%. This is indeed a valid expression for calculating the appreciated value.
Based on this analysis, the expressions that could be used to calculate the value of the house after [tex]\(t\)[/tex] years are:
- [tex]\(156,000 \cdot (1.12)^t\)[/tex]
- [tex]\(156,000 \cdot (1+0.12)^t\)[/tex]
These expressions account for the annual appreciation of 12%.
