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Answer :
To solve this problem, we need to determine which equation can help us calculate the amount of money to invest today so that it grows to [tex]$10,000 in 15 years, given an average yearly growth factor of 1.09.
1. Understand the problem:
- We want to have $[/tex]10,000 in the future (after 15 years).
- The annual growth factor of the investment is 1.09, meaning each year the investment grows by 9%.
2. We use the formula for compound interest, which is:
[tex]\[
\text{Future Value} = \text{Present Value} \times (\text{Growth Factor})^{\text{Years}}
\][/tex]
In this context:
- Future Value = $10,000
- Growth Factor = 1.09
- Years = 15
- Present Value = x (the amount we need to find)
3. Plug the known values into the compound interest formula:
[tex]\[
10000 = x \times (1.09)^{15}
\][/tex]
4. We need to solve for x, which represents the Present Value or the amount to invest today:
- Rearrange the equation to solve for x:
[tex]\[
x = \frac{10000}{(1.09)^{15}}
\][/tex]
5. Match this equation with the given options:
- The equation that matches our derived formula is:
[tex]\[
10000 = x(1.09)^{15}
\][/tex]
- This corresponds to option D.
Therefore, the correct equation to calculate the initial investment needed is option D: [tex]\(10000 = x(1.09)^{15}\)[/tex].
1. Understand the problem:
- We want to have $[/tex]10,000 in the future (after 15 years).
- The annual growth factor of the investment is 1.09, meaning each year the investment grows by 9%.
2. We use the formula for compound interest, which is:
[tex]\[
\text{Future Value} = \text{Present Value} \times (\text{Growth Factor})^{\text{Years}}
\][/tex]
In this context:
- Future Value = $10,000
- Growth Factor = 1.09
- Years = 15
- Present Value = x (the amount we need to find)
3. Plug the known values into the compound interest formula:
[tex]\[
10000 = x \times (1.09)^{15}
\][/tex]
4. We need to solve for x, which represents the Present Value or the amount to invest today:
- Rearrange the equation to solve for x:
[tex]\[
x = \frac{10000}{(1.09)^{15}}
\][/tex]
5. Match this equation with the given options:
- The equation that matches our derived formula is:
[tex]\[
10000 = x(1.09)^{15}
\][/tex]
- This corresponds to option D.
Therefore, the correct equation to calculate the initial investment needed is option D: [tex]\(10000 = x(1.09)^{15}\)[/tex].
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