To find the amount owed after 6 years when a principal amount of [tex]$49,000 is borrowed at an annual interest rate of 4% compounded annually, we will use the compound interest formula. Here is a step-by-step breakdown of the solution:
1. Identify the variables needed for the compound interest formula:
- Principal (P) = $[/tex]49,000
- Annual interest rate (r) = 4% or 0.04 (as a decimal)
- Number of years (t) = 6
2. The compound interest formula is:
[tex]\[
A = P \left(1 + r\right)^t
\][/tex]
where [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest.
3. Substitute the given values into the formula:
[tex]\[
A = 49000 \left(1 + 0.04\right)^6
\][/tex]
4. Calculate the growth factor:
[tex]\[
1 + 0.04 = 1.04
\][/tex]
5. Raise the growth factor to the power of the number of years:
[tex]\[
1.04^6
\][/tex]
6. Multiply the principal amount by the resultant value:
[tex]\[
A = 49000 \times 1.04^6
\][/tex]
7. Perform the multiplication to find the accumulated amount:
After performing the calculation (using a calculator for efficiency and accuracy):
[tex]\[
A \approx 62001
\][/tex]
8. Final result:
The amount owed after 6 years, rounded to the nearest dollar, is [tex]$62,001.
Thus, the amount owed after 6 years, when $[/tex]49,000 is borrowed at an annual interest rate of 4% compounded annually, is $62,001.
