Final answer:
To calculate the time needed for an investment to grow with compound interest, we use the formula for compound interest A = P(1 + r/n)^(nt) to solve for t. In this example, $4,000 grows to $17,000 with a monthly compounded interest rate of 6%, and the formula gives us the number of years when solved for t.
Step-by-step explanation:
To find how long it will take for $4,000 to grow to $17,000 with an interest rate of 6% compounded monthly, we can use the formula for compound interest.
A = P(1 + \( \frac{r}{n} \))^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.
In this case, P = $4,000, A = $17,000, r = 0.06, and n = 12.
To find t, we can rearrange the compound interest formula:
t = \( \frac{\log(\frac{A}{P})}{n \times \log(1 + \frac{r}{n})} \)
When we plug in the numbers:
t = \( \frac{\log(\frac{17000}{4000})}{12 \times \log(1 + \frac{0.06}{12})} \)
Calculating t gives us the number of years required for the investment to grow from $4,000 to $17,000.
Let's compute this using a calculator:
t = \( \frac{\log(4.25)}{12 \times \log(1 + \frac{0.06}{12})} \)
After the calculation, we round t to the nearest tenth of a year.