Question

Time Required to Reach a Goal

How many years will it take for an initial investment of \(10,000 to grow to \)25,000? Assume a rate of interest of 6% compounded continuously.

Short answer

If $10,000 is compounded continuously, it becomes $125,000 in approximately 15.271 years.

Step-by-step solution

Step 1. Given Information

Given to determine the number of years for an initial investment of $10,000 to grow to $25,000, assuming a rate of interest of 6% compounded continuously.

Step 2. Calculation for continuous compounding

According to the compound interest formula, the amount A after t years for the principal P with an annual rate of interest r compounded continuously is $A=P{e}^{rt}$

Here, the amount invested is $10,000 i.e. $P=10000$

Rate of interest is 6% i.e. $r=0.06$

Finally the amount becomes $25,000 i.e. $A=25000$

Plugging the values:

$$\begin{gathered} A=P{e}^{rt} \\ 25000=10000e^{0.06t} \\ {e}^{0.06t}=\frac{25000}{10000} \\ 0.06t=\ln 2.5 \\ t=\frac{\ln 2.5}{0.06} \\ t\cong 15.271 \end{gathered}$$

Step 3. Conclusion

Hence, if the amount is compounded continuously, it becomes $25,000 in approximately 15.271 years.