Diego and Layla deposit \( \$ 964.00 \) into a savings account which earns \( 14.6 \% \) interest compounded quarterly. They want to use the money in the account to go on a trip in 3 years How much will they be able to spend? 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. \( \$ \square \)

To find out how much Diego and Layla will be able to spend after 3 years, we need to calculate the balance in their savings account after 3 years using the formula for compound interest: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \] Given: - Principal (starting amount) \( P = \$964.00 \) - Interest rate \( r = 14.6\% = 0.146 \) (as a decimal) - Compounded quarterly, so \( n = 4 \) times per year - Time in years \( t = 3 \) years Substitute the given values into the formula: \[ A = 964.00\left(1 + \frac{0.146}{4}\right)^{4 \times 3} \] Now, we can calculate the balance in the savings account after 3 years. Calculate the value by following steps: - step0: Calculate: \(964\left(1+\frac{0.146}{4}\right)^{4\times 3}\) - step1: Divide the terms: \(964\left(1+\frac{73}{2000}\right)^{4\times 3}\) - step2: Add the numbers: \(964\left(\frac{2073}{2000}\right)^{4\times 3}\) - step3: Multiply the numbers: \(964\left(\frac{2073}{2000}\right)^{12}\) - step4: Simplify: \(964\times \frac{2073^{12}}{2000^{12}}\) - step5: Rewrite the expression: \(4\times 241\times \frac{2073^{12}}{2000^{12}}\) - step6: Rewrite the expression: \(4\times 241\times \frac{2073^{12}}{16^{12}\times 125^{12}}\) - step7: Rewrite the expression: \(4\times 241\times \frac{2073^{12}}{4^{24}\times 125^{12}}\) - step8: Reduce the numbers: \(241\times \frac{2073^{12}}{4^{23}\times 125^{12}}\) - step9: Multiply the fractions: \(\frac{241\times 2073^{12}}{4^{23}\times 125^{12}}\) The balance in the savings account after 3 years will be approximately $1482.21. Therefore, Diego and Layla will be able to spend $1482.21 on their trip.