Answer :
To determine which expressions could be used to calculate the future value of a house after appreciating annually at a rate of 12%, let's go through each option:
1. Option A: [tex]\( 156,000 \cdot (0.12)^t \)[/tex]
- This expression represents the house's value if it's being multiplied by 12% itself for [tex]\( t \)[/tex] years. Since 0.12 represents the appreciation rate, not the total increase factor, this doesn't model the correct increase of value in the neighborhood.
2. Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex]
- This expression adds the 12% to the original 100% of the house value, meaning that each year the house's value is multiplied by 1.12. Therefore, this correctly accounts for the fact that the house appreciates by 12% each year.
3. Option C: [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
- This is equivalent to Option B since [tex]\( 1 + 0.12 = 1.12 \)[/tex], so it correctly expresses the compound appreciation of 12% per year.
4. Option D: [tex]\( 156,000 \cdot (1 - 0.12)^t \)[/tex]
- This represents depreciation, where 12% is subtracted from 100% each year, which is not what the problem describes.
5. Option E: [tex]\( 156,000 \cdot \left(1 + \frac{0.12}{12}\right)^t \)[/tex]
- This expression seems to suggest monthly compounding of a 12% annual rate by dividing the rate by 12. However, since the problem states that the appreciation is annual, this option is mistakenly set up for monthly compounding for yearly period input and is not consistent with the annual appreciation scenario.
Thus, the correct expressions to use for calculating the appreciated future value of the house are:
- Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex]
- Option C: [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
These options correctly reflect an annual increase of 12%.
1. Option A: [tex]\( 156,000 \cdot (0.12)^t \)[/tex]
- This expression represents the house's value if it's being multiplied by 12% itself for [tex]\( t \)[/tex] years. Since 0.12 represents the appreciation rate, not the total increase factor, this doesn't model the correct increase of value in the neighborhood.
2. Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex]
- This expression adds the 12% to the original 100% of the house value, meaning that each year the house's value is multiplied by 1.12. Therefore, this correctly accounts for the fact that the house appreciates by 12% each year.
3. Option C: [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
- This is equivalent to Option B since [tex]\( 1 + 0.12 = 1.12 \)[/tex], so it correctly expresses the compound appreciation of 12% per year.
4. Option D: [tex]\( 156,000 \cdot (1 - 0.12)^t \)[/tex]
- This represents depreciation, where 12% is subtracted from 100% each year, which is not what the problem describes.
5. Option E: [tex]\( 156,000 \cdot \left(1 + \frac{0.12}{12}\right)^t \)[/tex]
- This expression seems to suggest monthly compounding of a 12% annual rate by dividing the rate by 12. However, since the problem states that the appreciation is annual, this option is mistakenly set up for monthly compounding for yearly period input and is not consistent with the annual appreciation scenario.
Thus, the correct expressions to use for calculating the appreciated future value of the house are:
- Option B: [tex]\( 156,000 \cdot (1.12)^t \)[/tex]
- Option C: [tex]\( 156,000 \cdot (1 + 0.12)^t \)[/tex]
These options correctly reflect an annual increase of 12%.
