Final answer:
The deposit needed by the student's uncle must be calculated using the present value of an annuity formula. After the values are inserted ($10,000 per year for 4 years at 8.5% interest), the deposit amount is obtained, and the remaining balances after the first and last withdrawals can be determined. Following the final withdrawal, the account balance will be $0.
Step-by-step explanation:
To determine the total deposit that the student's uncle must make to cover 4 years of education expenses at $10,000 per year with an 8.5% interest rate, we can use the present value of an annuity formula. For an annuity that makes n payments of P dollars, each one-year apart, starting one year from today, with an interest rate r, the present value PV is given by:
PV = P * ((1 - (1 + r)^-n) / r)
Plugging in the values:
P =$10,000
r = 8.5% or 0.085
n = 4
PV = $10,000 * ((1 - (1 + 0.085)^-4) / 0.085)
This gives us the deposit needed today. We can now calculate the amount left immediately after the first and last withdrawal using the remaining balance formula which subtracts the annuity payment from the future value of the present value:
Future Value after 1st withdrawal = PV * (1 + r) - P
Future Value after last withdrawal = PV * (1 + r)^n - P * (((1 + r)^n - 1) / r)
However, immediately after the last withdrawal, the future value will be $0 since all payments will have been made.