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Find out how long it takes a \[tex]$2700 investment to earn \$[/tex]500 interest if it is invested at 8% compounded quarterly. Round to the nearest tenth of a year. Use the formula [tex]A=P\left(1+\frac{r}{n}\right)^{nt}[/tex].

A) 2.5 years
B) 2.1 years
C) 1.9 years
D) 2.3 years


Sagot :

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].