https://file.aiquickdraw.com/mathbot/file/172608845007184blm0vo.jpg

The question asks to find out how long it will take for an initial capital \( P_0 = 20,000 \) invested at an annual compound interest rate of 5% to triple. The relevant logarithmic values are provided:

- \( \log(3) \approx 0.47 \)
- \( \log(1.05) \approx 0.02 \)

We can use the compound interest formula to model this situation:

\[
P = P_0(1 + r)^t
\]

where:
- \( P \) is the final amount,
- \( P_0 \) is the initial capital,
- \( r \) is the annual interest rate (in this case, 5%, or 0.05),
- \( t \) is the time in years.

Since we want the capital to triple, \( P = 3P_0 \), so the equation becomes:

\[
3P_0 = P_0(1.05)^t
\]

Dividing both sides by \( P_0 \):

\[
3 = (1.05)^t
\]

Now, take the logarithm of both sides:

\[
\log(3) = t \log(1.05)
\]

Substitute the provided values for \( \log(3) \approx 0.47 \) and \( \log(1.05) \approx 0.02 \):

\[
0.47 = t \times 0.02
\]

Solving for \( t \):

\[
t = \frac{0.47}{0.02} = 23.5 \text{ years}
\]

So, it will take approximately **23.5 years** for the capital to triple.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What is the formula for compound interest in continuous compounding?
2. How would the time change if the interest rate was higher, say 7%?
3. What would the time be if the capital was to quadruple instead of triple?
4. How does the approximation of logarithms affect the accuracy of the result?
5. How is the compound interest formula derived from basic principles?

**Tip**: When solving exponential growth problems like this one, logarithms are very useful for determining the time required for a certain level of growth.

Un capital inicial de P₀ = 20,000 es invertido a una tasa de interés compuesto del 5% anual. ¿Cuánto tiempo, en años, tardará el capital en triplicarse?

Solve how long it takes for an initial investment of P₀ = 20,000 to triple at a 5% annual compound interest rate using logarithms and exponential growth. Includes step-by-step solution using the compound interest formula and logarithmic calculations. Ideal for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation