How much money will there be in an account at the end of 6 years if \( \$ 14000 \) is deposited at \( 3 \% \) interest compounded quarterly? (Assume no withdrawals are made.)

To calculate the amount of money in an account after a certain period with compound interest, we can use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. Given: - \( P = 14000 \) - \( r = 0.03 \) (3% as a decimal) - \( n = 4 \) (since the interest is compounded quarterly) - \( t = 6 \) Now, we can substitute these values into the formula: \[ A = 14000 \left(1 + \frac{0.03}{4}\right)^{4 \times 6} \] Calculating \( \frac{0.03}{4} \): \[ \frac{0.03}{4} = 0.0075 \] Now, substituting this back into the formula: \[ A = 14000 \left(1 + 0.0075\right)^{24} \] Calculating \( 1 + 0.0075 \): \[ 1 + 0.0075 = 1.0075 \] Now we raise this to the power of 24: \[ A = 14000 \left(1.0075\right)^{24} \] Calculating \( (1.0075)^{24} \): \[ (1.0075)^{24} \approx 1.1914 \quad (\text{using a calculator}) \] Now, substituting this back into the equation for \( A \): \[ A \approx 14000 \times 1.1914 \approx 16678.00 \] Thus, the amount of money in the account at the end of 6 years will be approximately: \[ \boxed{16678.00} \]