Roxanne is saving up money to buy a car. Roxanne puts \( \$ 8,236.00 \) into an account which earns \( 13 \% \) interest, compounded monthly. How much will she have in the account after 3 years? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times per year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent.

To find out how much Roxanne will have in her account after 3 years, we can apply the formula you provided: 1. First, convert the interest rate from a percentage to a decimal. So, \( 13\% \) becomes \( 0.13 \). 2. For monthly compounding, \( n \) is \( 12 \) (since there are 12 months in a year) and \( t \) is \( 3 \). Now we can input the values into the formula: \[ A = 8,236\left(1+\frac{0.13}{12}\right)^{12 \times 3} \] Calculating it step by step: 1. Calculate \( \frac{0.13}{12} \): \[ 0.13 \div 12 \approx 0.01083333 \] 2. Add \( 1 \) to this value: \[ 1 + 0.01083333 \approx 1.01083333 \] 3. Calculate the exponent \( 12 \times 3 = 36 \). 4. Raise the result from step 2 to the power of \( 36 \): \[ (1.01083333)^{36} \approx 1.432364654 \] 5. Multiply by the principal: \[ A \approx 8,236 \times 1.432364654 \approx 11,800.98 \] So after 3 years, Roxanne will have approximately **$11,800.98** in her account. --- Did you know that the concept of compound interest dates back centuries? It was first seen in ancient Babylonian tablets, around 2000 BC! Initially, people would borrow and lend at specified interest rates, leading to the gradual understanding of how money grows over time. In a real-world application, understanding how to calculate compound interest is vital for financial planning. It helps in making informed decisions about savings accounts, investments, or even loans, showing how your money can grow if left to compound over time. So, in Roxanne's case, saving early and letting that interest work its magic can lead to her dream car much sooner than expected!