Exponential and Logarithmic Functions
Finding the final amount in a word problem on compound interest
Leanna
Laura deposited $\$ 4000$ into an account with $5.3 \%$ interest, compounded quarterly. Assuming that no withdrawals are made, how much will she have in the account after 9 years?

Do not round any intermediate computations, and round your answer to the nearest cent.
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Step 1 ：Laura deposited $4000 into an account with an annual interest rate of 5.3%, compounded quarterly. We are asked to find out how much she will have in the account after 9 years, assuming that no withdrawals are made.

Step 2 ：We can use the formula for compound interest to solve this problem. The formula is \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where: \n\n- \(A\) is the amount of money accumulated after n years, including interest. \n- \(P\) is the principal amount (the initial amount of money). \n- \(r\) is the annual interest rate (in decimal). \n- \(n\) is the number of times that interest is compounded per year. \n- \(t\) is the time the money is invested for, in years.

Step 3 ：Substituting the given values into the formula, we get: \n\n\(A = 4000 \left(1 + \frac{0.053}{4}\right)^{4 \times 9}\)

Step 4 ：Solving the equation, we find that \(A = 6424.776704834617\)

Step 5 ：Rounding to the nearest cent, we get the final amount as \$6424.78

Step 6 ：Final Answer: Laura will have approximately \$\boxed{6424.78} in the account after 9 years.