Answer :
To determine how much you should invest today to have [tex]$20,000 in 13 years, given an average yearly growth factor of 1.08, you can use the formula for calculating the present value of an investment. This formula rearranges the future value formula:
\[ \text{Future Value} = \text{Present Value} \times (\text{Growth Factor})^{\text{Number of Years}} \]
In this context:
- Future Value (FV) is $[/tex]20,000.
- Growth Factor (r) is 1.08.
- Number of Years (n) is 13.
- Present Value (PV) is the amount we want to find.
The formula to calculate the present value, given the future value, is:
[tex]\[ \text{Present Value} = \frac{\text{Future Value}}{(\text{Growth Factor})^{\text{Number of Years}}} \][/tex]
Using the provided numbers, we plug them into the formula:
[tex]\[ \text{Present Value} = \frac{20000}{(1.08)^{13}} \][/tex]
After evaluating the calculation, the present value turns out to be approximately [tex]$7353.96.
This means you should invest about $[/tex]7353.96 today to reach your goal of $20,000 in 13 years, given the growth factor. Therefore, the equation that should be used to calculate how much you need to invest today is:
[tex]\[ 20000 = x(1.08)^{13} \][/tex]
This matches the first option in the provided choices.
\[ \text{Future Value} = \text{Present Value} \times (\text{Growth Factor})^{\text{Number of Years}} \]
In this context:
- Future Value (FV) is $[/tex]20,000.
- Growth Factor (r) is 1.08.
- Number of Years (n) is 13.
- Present Value (PV) is the amount we want to find.
The formula to calculate the present value, given the future value, is:
[tex]\[ \text{Present Value} = \frac{\text{Future Value}}{(\text{Growth Factor})^{\text{Number of Years}}} \][/tex]
Using the provided numbers, we plug them into the formula:
[tex]\[ \text{Present Value} = \frac{20000}{(1.08)^{13}} \][/tex]
After evaluating the calculation, the present value turns out to be approximately [tex]$7353.96.
This means you should invest about $[/tex]7353.96 today to reach your goal of $20,000 in 13 years, given the growth factor. Therefore, the equation that should be used to calculate how much you need to invest today is:
[tex]\[ 20000 = x(1.08)^{13} \][/tex]
This matches the first option in the provided choices.
