To solve the problem of finding the amount owed when [tex]$1200 is borrowed for six years at an interest rate of 5% per year, compounded continuously, we use the formula for continuously compounded interest. Here are the detailed steps:
1. Understand the formula for continuously compounded interest:
The formula is:
\[ A = P \cdot e^{(r \cdot t)} \]
where:
- \( A \) is the amount owed,
- \( P \) is the principal amount (initial amount borrowed),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time the money is borrowed for (in years),
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
2. Identify the given values:
From the problem:
- The principal amount, \( P \), is $[/tex]1200.
- The annual interest rate, [tex]\( r \)[/tex], is 5%, which as a decimal is 0.05.
- The time, [tex]\( t \)[/tex], is 6 years.
3. Substitute the given values into the formula:
[tex]\[ A = 1200 \cdot e^{(0.05 \cdot 6)} \][/tex]
4. Calculate the exponent:
First, compute the product of the interest rate and time:
[tex]\[ 0.05 \cdot 6 = 0.30 \][/tex]
So we have:
[tex]\[ A = 1200 \cdot e^{0.30} \][/tex]
5. Use the value of [tex]\( e \)[/tex] raised to the power of 0.30:
The value of [tex]\( e^{0.30} \)[/tex] is approximately 1.34986. When we multiply this by the principal amount:
[tex]\[ A = 1200 \cdot 1.34986 \][/tex]
6. Calculate the amount owed:
[tex]\[ A \approx 1619.8305690912039 \][/tex]
7. Round the result to the nearest cent:
The exact amount after continuous compounding over six years ends up being approximately [tex]$1619.8305690912039. When rounded to the nearest cent, the final amount owed is:
\[ A \approx 1619.83 \]
Therefore, the amount owed, rounded to the nearest cent, is $[/tex]1619.83.
