Answer:
To find the total value of the prize, we need to calculate the sum of the weekly payments over 40 years. Each week, the prize pays $800, so the annual payment is $800 * 52 weeks = $41,600.
To calculate the total value of the prize over 40 years, we multiply the annual payment by the number of years:
Total value = Annual payment * Number of years
Total value = $41,600 * 40 years
Total value = $1,664,000
Therefore, the total value of the prize over 40 years is $1,664,000.
Now, to determine how much money the state needs to put into an account now to cover the weekly prize payments, we need to consider the interest earned on investments. If the state can earn 5% interest on investments, we can use the present value formula to calculate the initial investment needed.
The present value formula is:
Present value = Future value / (1 + interest rate)^n
Where:
Future value = Total value of the prize ($1,664,000)
Interest rate = 5% or 0.05
n = Number of years (40 years)
Calculating the present value:
Present value = $1,664,000 / (1 + 0.05)^40
Present value ≈ $425,678.97
Therefore, the state needs to put approximately $425,678.97 into an account now to cover the weekly prize payments over 40 years, considering a 5% interest rate on investments.