To solve the given equation,
[tex]\[ 13,967.08 = 11,000 \left(1 + \frac{0.04}{4}\right)^{4T} \][/tex]
we will find the value of [tex]\( T \)[/tex].
### Step-by-Step Solution:
1. Understand the equation:
The equation represents the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Here:
- [tex]\( A \)[/tex] is the future value, which is \[tex]$13,967.08.
- \( P \) is the principal amount (initial investment), which is \$[/tex]11,000.
- [tex]\( r \)[/tex] is the annual interest rate, 0.04 (or 4%).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year, which is 4 (quarterly).
- [tex]\( t \)[/tex] is the time the money is invested for in years, which we need to find.
2. Substitute the given values into the compound interest formula:
[tex]\[ 13,967.08 = 11,000 \left(1 + \frac{0.04}{4}\right)^{4T} \][/tex]
3. Simplify the expression inside the parenthesis:
[tex]\[ 1 + \frac{0.04}{4} = 1 + 0.01 = 1.01 \][/tex]
So, the equation becomes:
[tex]\[ 13,967.08 = 11,000 (1.01)^{4T} \][/tex]
4. Divide both sides by the principal amount [tex]\( 11,000 \)[/tex] to isolate the exponential term:
[tex]\[ \frac{13,967.08}{11,000} = (1.01)^{4T} \][/tex]
[tex]\[ 1.269734545 = (1.01)^{4T} \][/tex]
5. Take the natural logarithm (ln) of both sides to solve for the exponent:
[tex]\[ \ln(1.269734545) = \ln\left((1.01)^{4T}\right) \][/tex]
Using the logarithm power rule [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex], this becomes:
[tex]\[ \ln(1.269734545) = 4T \cdot \ln(1.01) \][/tex]
6. Calculate the natural logarithms:
- [tex]\(\ln(1.269734545) \approx 0.23880785929578607\)[/tex]
- [tex]\(\ln(1.01) \approx 0.009950330853168092\)[/tex]
The equation now is:
[tex]\[ 0.23880785929578607 = 4T \cdot 0.009950330853168092 \][/tex]
7. Divide both sides by [tex]\( 0.009950330853168092 \cdot 4 \)[/tex] to solve for [tex]\( T \)[/tex]:
[tex]\[ T = \frac{0.23880785929578607}{4 \cdot 0.009950330853168092} \][/tex]
The denominator [tex]\( 4 \cdot 0.009950330853168092 \approx 0.03980132341267237 \)[/tex]
[tex]\[ T = \frac{0.23880785929578607}{0.03980132341267237} \approx 5.999997960363094 \][/tex]
8. Interpret the result:
The value of [tex]\( T \)[/tex] is approximately [tex]\( 6 \)[/tex] years.
So, the investment will take approximately 6 years to grow from \[tex]$11,000 to \$[/tex]13,967.08 at an annual interest rate of 4%, compounded quarterly.
