To determine how long it takes for a \[tex]$2700 investment to earn \$[/tex]500 interest with an 8% annual interest rate compounded quarterly, we will use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Here:
- [tex]\( P \)[/tex] is the principal amount, \[tex]$2700.
- \( r \) is the annual interest rate, 8% or 0.08.
- \( n \) is the number of times the interest is compounded per year, 4 (since it is compounded quarterly).
- \( t \) is the time in years we need to find.
- \( A \) is the final amount after interest is added.
First, calculate the final amount \( A \). Since the investment earns \$[/tex]500 interest, we have:
[tex]\[ A = P + \$500 = 2700 + 500 = \$3200 \][/tex]
Now we can set up the equation:
[tex]\[ 3200 = 2700 \left(1 + \frac{0.08}{4}\right)^{4t} \][/tex]
Simplify the term inside the parentheses:
[tex]\[ 1 + \frac{0.08}{4} = 1 + 0.02 = 1.02 \][/tex]
Thus, our equation becomes:
[tex]\[ 3200 = 2700 (1.02)^{4t} \][/tex]
To isolate the exponent, divide both sides by 2700:
[tex]\[ \frac{3200}{2700} = (1.02)^{4t} \][/tex]
[tex]\[ \frac{3200}{2700} = 1.185185 \][/tex]
Therefore:
[tex]\[ 1.185185 = (1.02)^{4t} \][/tex]
To solve for [tex]\( t \)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(1.185185) = \ln((1.02)^{4t}) \][/tex]
Using the logarithm power rule [tex]\( \ln(a^b) = b \ln(a) \)[/tex]:
[tex]\[ \ln(1.185185) = 4t \ln(1.02) \][/tex]
Now, solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(1.185185)}{4 \ln(1.02)} \][/tex]
Using the given answer:
[tex]\[ t \approx 2.14490524734784 \][/tex]
Rounding to the nearest tenth:
[tex]\[ t \approx 2.1 \][/tex]
So, it takes approximately 2.1 years for a \[tex]$2700 investment to earn \$[/tex]500 interest at an 8% annual interest rate compounded quarterly.
The correct answer is [tex]\( \boxed{2.1} \)[/tex].
