Final answer:
To find out how long it will take for an investment to grow to a certain amount with compound interest, we use the compound interest formula and solve for the time variable. In this case, it takes approximately 15.22 years for a $3,000 investment with a 7% annual interest rate compounded quarterly to grow to $4,470.
Step-by-step explanation:
To solve the mathematical problem of calculating the time it takes for an investment to grow from $3,000 to $4,470 at a 7% annual interest rate compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)(nt)
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 (in decimal form).
n is the number of times that interest is compounded per year.
t is the time the money is invested for, in years.
We have:
A = $4,470
P = $3,000
r = 0.07 (7% in decimal form)
n = 4 (since compounded quarterly)
We need to find t and we can rearrange the formula to solve for t:
t = (log(A/P)) / (n*log(1 + r/n))
Substituting the given values, we get:
t = (log(4,470 / 3,000)) / (4 * log(1 + 0.07/4)
Calculating the values, we get:
t = (log(1.49)) / (4 * log(1.0175)
t ≈ 15.22
Therefore, it will take approximately 15.22 years for the investment to grow to $4,470.