Use the formula for continuously compounded interest, A = Pert, to find the annual interest rate for an $8000 investment that earns $410.17 in one year.

The formula for continuously compounded interest is A = Pe^rt, where:

A = the amount of money accumulated after n years, including interest. P = the principal amount (the initial amount of money) r = annual interest rate (in decimal) t = time the money is invested for in years

In this case, you know that:

A = $8000 +$410.17 = $8410.17 (This is the total amount after one year) P =$8000 (This is the initial amount) t = 1 year

You're trying to find 'r', the annual interest rate.

So you can rearrange the formula to solve for r:

r = ln(A/P) / t

Substitute the known values into the equation:

r = ln($8410.17/$8000) / 1

Now, calculate the natural logarithm and the division:

r ≈ 0.0500

To express this as a percentage, multiply by 100:

r ≈ 5.00%

So, the annual interest rate for an $8000 investment that earns$410.17 in one year is approximately 5.00%.