To determine the value of the principal investment [tex]\( P \)[/tex], we need to use the compound interest formula:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
Given:
- [tex]\( A \)[/tex] is the final amount, which is \[tex]$ 12,065.51
- \( r \) is the annual interest rate, which is 5% or 0.05
- \( n \) is the number of times the interest is compounded per year, which is 4 (quarterly)
- \( t \) is the time the money is invested in years, which is 15 years
The formula becomes:
\[ 12065.51 = P \left( 1 + \frac{0.05}{4} \right)^{4 \times 15} \]
First, we need to calculate the term inside the parenthesis:
\[ 1 + \frac{0.05}{4} = 1 + 0.0125 = 1.0125 \]
Next, raise this value to the power of \( 4t \) (which is \( 4 \times 15 = 60 \)):
\[ (1.0125)^{60} \]
Using a calculator or logarithmic tables, we find that:
\[ (1.0125)^{60} \approx 2.108665 \]
Now, rearrange the formula to solve for \( P \):
\[ P = \frac{A}{(1.0125)^{60}} \]
Substitute \( A \) and the calculated value:
\[ P = \frac{12065.51}{2.108665} \]
\[ P \approx 5725.90015446792 \]
Rounding this to the nearest hundredths place, we get:
\[ P \approx 5725.90 \]
Therefore, the value of the principal investment is \$[/tex] 5725.90. The correct answer is:
[tex]\[ \boxed{5725.90} \][/tex]
