A person invests $\$ 100$, which earns interest that is compounded yearly.
This graph represents the value of this investment over time.
Approximately how many years does it take for the investment to double in value?
2 years 12 years
14 years 200 years
Answer
\(\boxed{\text{It takes approximately 14 years for the investment to double in value.}}\)
Step 1 ：Given the initial investment (P) is $100 and we want to find out when it doubles, so the future value (A) is $200. We assume a reasonable interest rate of 5% (0.05) and the interest is compounded yearly (n=1).
Step 2 ：Use the formula for compound interest: \(A = P(1 + \frac{r}{n})^{nt}\)
Step 3 ：Plug in the values: \(200 = 100(1 + \frac{0.05}{1})^{1*t}\)
Step 4 ：Solve for t: \(t = \frac{\log{\frac{200}{100}}}{\log{1.05}} \approx 14.21\)
Step 5 ：\(\boxed{\text{It takes approximately 14 years for the investment to double in value.}}\)
