Sure! Let's solve this step-by-step.
1. Identify the Parameters:
- Initial investment, [tex]\( P = \$100,000 \)[/tex]
- Future value, [tex]\( A = \$1,000,000 \)[/tex]
- Interest rate, [tex]\( r = 7.5\% \)[/tex] or [tex]\( 0.075 \)[/tex]
- We are using the formula for continuous compounding.
2. Use the Continuous Compounding Formula:
The formula for continuous compounding is:
[tex]\[
A = P e^{rt}
\][/tex]
3. Rearrange to Solve for [tex]\( t \)[/tex]:
We need to find the time [tex]\( t \)[/tex], so we rearrange the formula to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(A/P)}{r}
\][/tex]
4. Calculate the Natural Logarithm:
Substitute the values:
[tex]\[
t = \frac{\ln(1,000,000 / 100,000)}{0.075}
\][/tex]
Simplify inside the logarithm:
[tex]\[
t = \frac{\ln(10)}{0.075}
\][/tex]
5. Evaluate [tex]\( \ln(10) \)[/tex]:
The natural logarithm of 10 is approximately 2.302585.
6. Complete the Calculation:
[tex]\[
t = \frac{2.302585}{0.075}
\][/tex]
[tex]\[
t \approx 30.701134573253945
\][/tex]
7. Round to the Nearest Year:
The nearest integer value of 30.701134573253945 is 31.
Therefore, it will take approximately 31 years for the money to grow from \[tex]$100,000 to \$[/tex]1,000,000 at an interest rate of 7.5% compounded continuously.
Final Answer:
It will take approximately 31 years.
