To determine how much Melissa needs to invest to achieve a future value of \[tex]$114,700 after 20 years with an annual interest rate of 3.42% compounded daily, we will use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \(A\) is the future value of the investment, which is \$[/tex]114,700.
- [tex]\(P\)[/tex] is the principal amount (the initial amount to be invested).
- [tex]\(r\)[/tex] is the annual interest rate (in decimal form), which is 0.0342.
- [tex]\(n\)[/tex] is the number of times interest is compounded per year, which is 365 for daily compounding.
- [tex]\(t\)[/tex] is the number of years the money is invested, which is 20 years.
We need to rearrange the formula to solve for [tex]\(P\)[/tex]:
[tex]\[ P = \frac{A}{ \left(1 + \frac{r}{n}\right)^{nt} } \][/tex]
Let's substitute the given values into the formula and solve for [tex]\(P\)[/tex]:
1. Substitute the given values into the formula:
[tex]\[ P = \frac{114700}{ \left(1 + \frac{0.0342}{365}\right)^{365 \times 20} } \][/tex]
2. Compute the base of the exponent:
[tex]\[ 1 + \frac{0.0342}{365} = 1 + 0.00009369863 \approx 1.00009369863 \][/tex]
3. Compute the exponent:
[tex]\[ 365 \times 20 = 7300 \][/tex]
4. Now compute the full expression in the denominator:
[tex]\[ \left(1.00009369863\right)^{7300} \approx 1.981314167 \][/tex]
5. Now, plug this back into our formula for [tex]\(P\)[/tex]:
[tex]\[ P = \frac{114700}{1.981314167} \approx 57878.85197906927 \][/tex]
Finally, we round the result to the nearest dollar:
[tex]\[ P \approx 57879 \][/tex]
So, Melissa would need to invest \[tex]$57,879 to have \$[/tex]114,700 after 20 years with an annual interest rate of 3.42% compounded daily.
