Exponentsey and toguilthmic Functions
Finding the final amount in a word problem on continuous compound.
Nathanuel R.
An initial amount of $\$ 3200$ is invested in an account at an interest rate of $7.5 \%$ per year, compounded continuously. Assuming that no withdrawals are made, find the amount in the account after four years.
Do not round any intermediate computations, and round your answer to the nearest cent.
Answer
\(\boxed{A ≈ $4319.54}\)
Step 1 ：Given the principal amount (P) is $3200, the annual interest rate (r) is 7.5% or 0.075 in decimal, and the time (t) is 4 years.
Step 2 ：The formula for continuous compounding is \(A = Pe^{rt}\), where A is the amount of money accumulated after n years, including interest.
Step 3 ：Substitute the given values into the formula: \(A = 3200 * e^{(0.075*4)}\)
Step 4 ：Calculate the value of \(e^{(0.075*4)}\) first: \(e^{0.3} ≈ 1.349858807576003\)
Step 5 ：Then, multiply this value by the principal amount: \(A = 3200 * 1.349858807576003\)
Step 6 ：\(\boxed{A ≈ $4319.54}\)
