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 :\(\boxed{A ≈ $4319.54}\)