Carl purchased a rare painting for $13597.50.
The painting increased in value each year by 6.1% of the original price.
What was the value of the painting after 12 years?

To determine the value of the painting after 12 years, we will calculate the increase in value each year based on the original price and then sum these increases to the original price.

Given:
- Original price of the painting $ P_0 = \$13597.50 $
- Annual increase rate $ r = 6.1\% = 0.061 $

First, we calculate the annual increase in value:
\[ \text{Annual increase} = P_0 \times r = 13597.50 \times 0.061 = 829.4475 \]

Since the painting increases in value by this fixed amount every year, the total increase over 12 years will be:
\[ \text{Total increase over 12 years} = \text{Annual increase} \times 12 = 829.4475 \times 12 = 9953.37 \]

Now, we add this total increase to the original price to find the value after 12 years:
\[ \text{Value after 12 years} = P_0 + \text{Total increase over 12 years} = 13597.50 + 9953.37 = 23550.87 \]

Therefore, the value of the painting after 12 years is:
\[ \$23550.87 \]

Do you want more details or have any questions?

Here are some related questions:
1. What would be the value of the painting after 15 years with the same increase rate?
2. How would the value change if the annual increase rate was 7% instead of 6.1%?
3. What is the formula for calculating the value of the painting after any number of years?
4. How much total interest does the painting accrue over the 12 years?
5. How does the compounding of interest affect the final value of an investment compared to a simple interest increase?
6. Can the increase in value be modeled using exponential growth instead of linear growth? If so, what would be the formula?
7. What would be the value of the painting after 20 years using the same annual increase rate?
8. How does this problem change if the increase rate is applied to the new value each year (compounded annually) instead of the original price?

**Tip:** Understanding the difference between simple interest and compound interest is crucial for accurately modeling the growth of investments.

Explore how to calculate the value of a painting after 12 years with an annual increase rate of 6.1%. This problem involves understanding percentage increase and compound interest concepts, with detailed step-by-step solutions and related questions.

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