To determine the number of years it will take for an initial investment of [tex]$20,000 to grow to $[/tex]30,000 with a continuous compounding interest rate of 12%, follow these steps:
#### Step 1: Identify the given values.
- Initial investment ([tex]\(P\)[/tex]): [tex]$20,000
- Final amount (\(A\)): $[/tex]30,000
- Interest rate ([tex]\(r\)[/tex]): 12% (which is 0.12 in decimal form)
#### Step 2: Write down the formula for continuous compound interest.
The formula for continuous compound interest is:
[tex]\[ A = P \cdot e^{(rt)} \][/tex]
#### Step 3: Rearrange the formula to solve for time ([tex]\(t\)[/tex]).
We need to isolate [tex]\(t\)[/tex]. Start by dividing both sides by [tex]\(P\)[/tex]:
[tex]\[ \frac{A}{P} = e^{(rt)} \][/tex]
Next, take the natural logarithm (ln) of both sides to remove [tex]\(e\)[/tex]:
[tex]\[ \ln\left(\frac{A}{P}\right) = rt \][/tex]
Now, solve for [tex]\(t\)[/tex] by dividing both sides by [tex]\(r\)[/tex]:
[tex]\[ t = \frac{1}{r} \cdot \ln\left(\frac{A}{P}\right) \][/tex]
#### Step 4: Plug in the known values.
[tex]\[ t = \frac{1}{0.12} \cdot \ln\left(\frac{30,000}{20,000}\right) \][/tex]
#### Step 5: Simplify inside the natural logarithm function.
[tex]\[ \frac{30,000}{20,000} = 1.5 \][/tex]
So, the equation now is:
[tex]\[ t = \frac{1}{0.12} \cdot \ln(1.5) \][/tex]
#### Step 6: Calculate the natural logarithm of 1.5.
The natural logarithm of 1.5 is approximately 0.405465.
#### Step 7: Substitute the approximate value of [tex]\(\ln(1.5)\)[/tex] back into the equation and do the multiplication.
[tex]\[ t = \frac{1}{0.12} \cdot 0.405465 \][/tex]
[tex]\[ t = 8.3788759009013702 \][/tex]
#### Step 8: Finally, round the result to two decimal places.
[tex]\[ t \approx 3.38 \][/tex]
#### Conclusion
It will take about 3.38 years for the investment to grow to $30,000 with a continuous compounding interest rate of 12%.
