To solve this problem, we will use the compound interest formula and solve for the principal investment [tex]\( P \)[/tex]. The compound interest formula is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the final amount, which is [tex]\( 13,997.55 \)[/tex] dollars.
- [tex]\( r \)[/tex] is the annual interest rate, which is [tex]\( 7 \% \)[/tex] or [tex]\( 0.07 \)[/tex].
- [tex]\( n \)[/tex] is the number of times interest is compounded per year, which is [tex]\( 4 \)[/tex] (quarterly compounding).
- [tex]\( t \)[/tex] is the number of years, which is [tex]\( 15 \)[/tex] years.
First, we will rearrange the formula to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \][/tex]
Plugging in the given values:
1. Calculate [tex]\( \left(1 + \frac{r}{n}\right) \)[/tex]:
[tex]\[ 1 + \frac{0.07}{4} = 1 + 0.0175 = 1.0175 \][/tex]
2. Calculate the exponent [tex]\( nt \)[/tex]:
[tex]\[ n \cdot t = 4 \cdot 15 = 60 \][/tex]
3. Raise the base to the exponent:
[tex]\[ 1.0175^{60} \approx 2.8318162778223424 \][/tex]
4. Divide the final amount by this compound factor:
[tex]\[ P = \frac{13,997.55}{2.8318162778223424} \approx 4942.958379617788 \][/tex]
5. Round to the nearest hundredths place:
[tex]\[ P \approx 4942.96 \][/tex]
Thus, the value of the principal investment, rounded to the nearest hundredths place, is:
[tex]\[ \boxed{4,942.96} \][/tex]
