Final answer:
To find how many years it will take to accumulate $50,000 with yearly $8,000 deposits at a 5% annual interest rate, the future value of annuity formula is used, rearranged to solve for the number of years. The specific calculation involves logarithms after plugging the known values into the formula.
Step-by-step explanation:
The student's question pertains to compound interest and how it can be used to determine the time required for an investment to reach a certain goal. Specifically, we need to calculate the number of years it will take for yearly deposits of $8,000 to accumulate to $50,000 in a bank account with a 5% annual interest rate.
To solve this, we will use the formula for the future value of an annuity, which accounts for regular deposits and compound interest:
Future Value of Annuity (FVA) = Pmt * [(1 + r)^n - 1] / r
Where,
Pmt is the annual deposit ($8,000),
r is the annual interest rate (5% or 0.05),
n is the number of years,
and FVA is the future value we want to accumulate ($50,000).
We will rearrange the formula to solve for n, the number of years:
n = [log(FVA/Pmt + 1) / log(1 + r)] - 1
Inserting the known values and solving for n, we get:
n = [log($50,000/$8,000 + 1) / log(1 + 0.05)] - 1
When the calculations are performed, this will give us the precise number of years it will take to reach the $50,000 target with annual deposits of $8,000 at 5% interest.