Final answer:
The initial investment is calculated using the formula for compound interest. However, after solving the equation with the provided interest rate, compounding frequency, and total amount after 10 years, the result doesn't match any of the given options. It's possible there is an error in the figures provided.
Step-by-step explanation:
To find the initial investment for an account that has been compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)(n*t)
Where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested for in years.
We know that A = $3910.94, r = 4.5% or 0.045, n = 4 (since the interest is compounded quarterly), and t = 10 years. Plugging these values into the formula:
3910.94 = P(1 + 0.045/4)(4*10)
To solve for P, we divide both sides by (1 + 0.045/4)(4*10):
P = 3910.94 / (1 + 0.045/4)(4*10)
Calculating this gives us the initial investment.
Using a calculator:
P ≈ 3910.94 / (1.01125)40
P ≈ 3910.94 / 1.95614
P ≈ $1998.01
This result does not match any of the provided options. There might have been a mistake in the calculations or in the options provided. Can you please check the options or the details of the question again?