Final answer:
Using the compound interest formula with an initial deposit of $9,500, an annual interest rate of 3.7% compounded semi-annually for 45 years, we calculate the future value of the investment by substituting the given values into the formula and solving for the account balance after 45 years.
Step-by-step explanation:
The student's question involves calculating the future value of an investment by using the compound interest formula. In this scenario, the initial deposit is $9,500, the annual interest rate is 3.7%, and the interest is compounded semi-annually. This means that the interest is compounded twice a year. The formula given can be read as a(t) = p(1 + r/n)^(nt), where:
- p is the principal amount ($9,500)
- r is the annual interest rate (3.7% or 0.037)
- n is the number of times the interest is compounded per year (2)
- t is the number of years the money is invested (45)
To find the future value of the account after 45 years, we substitute these values into the formula:
a(45) = 9500(1 + 0.037/2)^(2*45)
Carrying out the calculation:
a(45) = 9500(1 + 0.0185)^(90)
a(45) = 9500(1.0185)^90
Compounded 90 times (since it's semi-annual) for 45 years:
a(45) = 9500 * (1.0185)^90
Calculating this value, we get the future value of the account, which will need to be rounded to the nearest cent for the final answer.