To determine the interest a company needs to pay after 3 years for borrowing [tex]$50,000 at an annual interest rate of 4% with interest compounded annually, we follow these steps:
1. Identify the given values:
- Principal amount (P) = $[/tex]50,000
- Annual interest rate (r) = 4% or 0.04 in decimal form
- Time (t) = 3 years
2. Use the compound interest formula to calculate the amount at the end of 3 years:
[tex]\[
A = P \left(1 + r\right)^t
\][/tex]
Where:
- [tex]\( A \)[/tex] is the amount after time [tex]\( t \)[/tex]
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( t \)[/tex] is the time in years
3. Substitute the given values into the formula:
[tex]\[
A = 50000 \left(1 + 0.04\right)^3
\][/tex]
4. Calculate each step:
[tex]\[
A = 50000 \left(1.04\right)^3
\][/tex]
5. Compute [tex]\( (1.04)^3 \)[/tex]:
[tex]\[
(1.04)^3 \approx 1.124864
\][/tex]
6. Multiply the principal amount by the result:
[tex]\[
A = 50000 \times 1.124864 \approx 56243.2
\][/tex]
7. Determine the interest alone by subtracting the principal amount from the total amount:
[tex]\[
\text{Interest} = A - P
\][/tex]
[tex]\[
\text{Interest} = 56243.2 - 50000 = 6243.2
\][/tex]
Therefore, the company needs to pay [tex]$6,243.20 in interest after 3 years. The correct option is:
$[/tex]6,243.20
