Step 1 :We are given that the final amount (A) is $7500, the interest rate (r) is 2.6% or 0.026 in decimal form, and the time (t) is 12 years. We need to find the initial investment (P).
Step 2 :The formula for continuously compounded interest is \(A = P * e^{rt}\). We can rearrange this formula to solve for P: \(P = \frac{A}{e^{rt}}\).
Step 3 :Substituting the given values into the formula, we get \(P = \frac{7500}{e^{0.026*12}}\).
Step 4 :Calculating the above expression, we find that the initial investment should be approximately $5490.
Step 5 :\(\boxed{5490}\) dollars should be the initial investment.