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To calculate the annual increase in the value of the comic book, we can use the compound interest formula:

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

Where:
- $A$ is the final amount ($786,000 in 2020),
- $P$ is the initial amount ($0.05 in 1939),
- $r$ is the annual growth rate (which we are solving for),
- $t$ is the number of years (2020 - 1939 = 81 years).

### Steps:
1. Rearrange the formula to solve for $r$:
\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

2. Substitute the values into the equation:
\[
r = \left(\frac{786,000}{0.05}\right)^{\frac{1}{81}} - 1
\]

3. Simplify the fraction:
\[
r = \left(15,720,000\right)^{\frac{1}{81}} - 1
\]

4. Calculate the value of $ \left(15,720,000\right)^{\frac{1}{81}} $.

Let me do this calculation:

\[
\left(15,720,000\right)^{\frac{1}{81}} \approx 1.2853
\]

5. Subtract 1 to find the annual increase rate:
\[
r \approx 1.2853 - 1 = 0.2853
\]

6. Convert the decimal to a percentage:
\[
r \times 100 \approx 28.53\%
\]

Thus, the annual increase in the value of the comic book is approximately **28.53%**.

Do you have any questions or want further details on this?

### Related Questions:
1. What if the comic book had been sold for a different amount in 2020?
2. How does the annual increase compare if the years were different?
3. What would happen to the rate if the initial price had been higher?
4. How does compound interest apply in other investment scenarios?
5. Can this formula be used for calculating depreciation instead of appreciation?

**Tip**: The compound interest formula can be applied in a variety of financial contexts, including loans, investments, and savings.

Assume that in 2020, the first edition of a comic book was sold at auction for $786,000. The comic book was originally sold in 1939 for $0.05. What was the annual increase in the value of the comic book? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places.)

Learn how to calculate the annual increase in the value of a comic book sold for $786,000 in 2020, originally priced at $0.05 in 1939. This example uses the compound interest formula to find the annual growth rate over 81 years. Perfect for students studying compound interest and exponential growth, suitable for Grades 10-12.

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