Final answer:
To find the time needed for Vijay's $4,000 investment to grow to $14,217 at 14% interest compounded monthly, we use the compound interest formula and solve for the time variable, t.
Step-by-step explanation:
The question involves determining the duration required for an initial investment to reach a certain amount through the process of compound interest. Given an initial investment of $4,000 with a 14% interest rate compounded monthly, the goal is to calculate the time needed for this investment to grow to $14,217 to start a business.
To solve this problem, we use the compound interest formula:
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.
We rearrange the formula to solve for t:
t = \(\frac{\log(A/P)}{n \cdot \log(1 + r/n)}\)
By plugging in the values:
A = $14,217, P = $4,000, r = 0.14, n = 12,
We can calculate the number of years t necessary for Vijay's investment to reach his desired amount.
Vijay's diligent savings and understanding of compound interest will help him reach his business goals.