https://file.aiquickdraw.com/mathbot/file/1727269433013qp142tsy.jpg

This problem is about continuous compounding of interest, and we can use the formula for continuous compounding:

\[
A = P e^{rt}
\]

where:
- $A$ is the final amount (in this case, double the investment),
- $P$ is the initial principal (in this case, $1000),
- $r$ is the annual interest rate (7%, or 0.07),
- $t$ is the time in years,
- $e$ is the base of the natural logarithm.

### Step 1: Set up the equation
We are asked how long it will take to double the investment, so:

\[
2P = P e^{0.07t}
\]

Simplifying the equation:

\[
2 = e^{0.07t}
\]

### Step 2: Solve for $t$
Take the natural logarithm on both sides:

\[
\ln(2) = 0.07t
\]

Now, solve for $t$:

\[
t = \frac{\ln(2)}{0.07}
\]

Let me compute this value.The time required to double the investment is approximately 9.90 years.

So, the correct answer is **C) 9.90 years**.

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. What would the time be if the interest rate were 5% instead of 7%?
2. How would the formula change if interest were compounded annually instead of continuously?
3. What is the doubling time for an investment at a 10% continuous interest rate?
4. What does the natural logarithm represent in the context of compound interest?
5. How can you estimate doubling time without logarithms (using the Rule of 72)?

**Tip:** The Rule of 72 is a quick way to estimate the doubling time of an investment by dividing 72 by the interest rate percentage.

An initial investment of $1000 grows at an annual interest rate of 7% compounded continuously. How long will it take to double the investment?

Discover how to calculate the time required to double an investment of $1000 at a 7% continuous compounding interest rate. This step-by-step solution covers exponential growth and logarithmic functions. Suitable for high school students in 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