Final answer:
To determine the future value of a savings account with 4% annual interest, compound interest calculations are used. After 3 years, the account will have approximately $1124.86, and after 16 years, $1809.48. It will take roughly 10.67 years to reach $1500, and about 17.67 years to accumulate $2000.
Step-by-step explanation:
The student is asking about the future value of a savings account with an interest rate of 4% compounded annually. To calculate the amount in the account after a certain number of years, the formula for compound interest is used:
A = P(1 + r/n)^(nt)
Where:
- A = the future value of the investment/loan, including interest
- P = the principal investment amount (initial deposit)
- r = the annual interest rate (decimal)
- n = the number of times that interest is compounded per year
- t = the number of years the money is invested or borrowed for
Assuming the interest is compounded once a year (n = 1), the following calculations can be made:
a. Amount after 3 years:
A = $1000 * (1 + 0.04/1)^(1*3) = $1000 * (1.04)^3 = $1124.86
b. Amount after 16 years:
A = $1000 * (1.04)^16 = $1809.48
c. Years to contain $1500:
1500 = $1000 * (1.04)^t
t = ln(1500/1000) / ln(1.04) ≈ 10.67 years
d. Years to contain $2000:
2000 = $1000 * (1.04)^t
t = ln(2000/1000) / ln(1.04) ≈ 17.67 years