Final answer:
To find out how long it takes for $15,000 to grow to $50,000 at 6.5% annual interest compounded annually, we use the compound interest formula. After substituting the values and solving for time, we discover it takes approximately 22.5 years.
Step-by-step explanation:
To calculate the time needed for an investment to grow from $15,000 to $50,000 at an annual interest rate of 6.5%, compounded annually, we 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 (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
Given that the interest is compounded annually, n = 1. We want to find t when A = $50,000, P = $15,000, and r = 0.065 (6.5%).
Substituting the values into the formula and solving for t, we get:
$50,000 = $15,000(1 + 0.065)t
We can rewrite this equation as:
3.3333 = (1.065)t
Using logarithms to solve for t:
t = log(3.3333) / log(1.065)
Calculating this gives us t ≈ 22.5 years.
Therefore, it would take approximately 22.5 years for the investment to grow to $50,000, making the correct answer b. 22.5 years.