To solve the problem of finding the value of Haley's mutual fund investment in 15 years, we'll use the following exponential function formula for compound interest:
[tex]\[ A = P(1 + r)^t \][/tex]
where:
- [tex]\(A\)[/tex] is the amount of money accumulated after [tex]\(t\)[/tex] years, including interest.
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money).
- [tex]\(r\)[/tex] is the annual interest rate (decimal).
- [tex]\(t\)[/tex] is the number of years the money is invested.
Step-by-Step Solution:
1. Identify the given values:
- Principal amount ([tex]\(P\)[/tex]): [tex]$750
- Annual interest rate (\(r\)): 3.5% (which is 0.035 in decimal form)
- Number of years (\(t\)): 15
2. Substitute these values into the formula:
\[
A = 750 \times (1 + 0.035)^{15}
\]
3. Compute the expression inside the parenthesis:
\[
1 + 0.035 = 1.035
\]
4. Raise this result to the power of 15:
\[
1.035^{15}
\]
5. Multiply this result by the principal amount \(750\):
\[
750 \times 1.035^{15} \approx 1256.51
\]
So the value of the mutual fund after 15 years is approximately $[/tex]1256.51.
Therefore, the number you would fill in for [tex]\(a\)[/tex] to solve the equation [tex]\(3 = 8\)[/tex] is outside the scope of this problem since it concerns a different equation. For the given investment problem, the initial principal (a) is [tex]$750.
The final answer is approximately $[/tex]1256.51, indicating the value of Haley's mutual fund in 15 years.
