To determine the value of [tex]\( P \)[/tex] (the initial principal) given that the account balance [tex]\( A \)[/tex] after 7 years is \$10,478, we can use the compound interest formula.
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for in years.
Given:
- [tex]\( A = 10,478 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( n = 4 \)[/tex] (quarterly compounding)
- [tex]\( t = 7 \)[/tex] years
We need to solve for [tex]\( P \)[/tex].
First, compute the effective interest rate per period:
[tex]\[ \text{Effective rate per period} = 1 + \frac{r}{n} = 1 + \frac{0.03}{4} = 1 + 0.0075 = 1.0075 \][/tex]
Next, compute the total number of compounding periods:
[tex]\[ nt = n \times t = 4 \times 7 = 28 \][/tex]
Now, raise the effective rate to the power of the total number of compounding periods:
[tex]\[ \text{Accumulated rate} = (1.0075)^{28} \][/tex]
Given [tex]\( A \)[/tex], we can now solve for [tex]\( P \)[/tex] by rearranging the compound interest formula:
[tex]\[ P = \frac{A}{\text{Accumulated rate}} = \frac{10,478}{(1.0075)^{28}} \][/tex]
After calculating, we find:
[tex]\[ P \approx 8,499.96 \][/tex]
Therefore, the initial principal [tex]\( P \)[/tex] rounded to the nearest hundredths place is:
[tex]\[ \boxed{8499.96} \][/tex]
Since this value does not match any of the given multiple-choice options, but it seems the closest correct value would be if we considered rounding in different contexts. Since we explicitly rounded to the correct nearest hundredths place, the best answer would be:
[tex]\( P = 8,500.00 \)[/tex]
