Final answer:
To have $10,000 in ten years at a 10% annual compound interest rate, Zoe would need to invest approximately $3,855.43 today. This is calculated using the compound interest formula and rearranging it to solve for the initial investment required.
Step-by-step explanation:
When calculating how much money Zoe needs to invest today to have $10,000 in ten years at a 10% annual compound interest rate, we can use the formula for compound interest: A = P(1+r/n)nt, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
Since we're looking for the initial investment (P), we can rearrange the formula to solve for P: P = A / (1+r/n)nt. We're given that A = $10,000, r = 0.10 (10%), n = 1 (compounded annually), and t = 10 years.
Plugging in the values, we get: P = $10,000 / (1+0.10/1)1*10 = $10,000 / (1.10)10. Calculating the denominator, we get (1.10)10 ≈ 2.59374. Therefore, P = $10,000 / 2.59374 ≈ $3,855.43. So, Zoe would need to invest approximately $3,855.43 today at a 10% interest rate compounded annually to have $10,000 in ten years.