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The problem provides the value of a painting as a function of time $ t $ (in years) given by:

\[
V(t) = 25000 \cdot e^{0.14t}
\]

Let's solve each part step by step.

### (a) Find the value of the painting when the woman bought it.

When the woman bought the painting, $ t = 0 $. Substitute $ t = 0 $ into the equation:

\[
V(0) = 25000 \cdot e^{0.14 \cdot 0} = 25000 \cdot e^0 = 25000 \cdot 1 = 25000
\]

So, the value of the painting when the woman bought it is **$25,000**.

### (b) Find the value of the painting after 9 years.

Now, substitute $ t = 9 $ into the equation:

\[
V(9) = 25000 \cdot e^{0.14 \cdot 9} = 25000 \cdot e^{1.26}
\]

Let's calculate the numerical value.

\[
V(9) \approx 25000 \cdot e^{1.26} \approx 25000 \cdot 3.524 \approx 88100
\]

So, the value of the painting after 9 years is approximately **$88,100**.

### (c) Find the difference in the value of the painting after 10 years compared to after 9 years.

First, calculate the value of the painting after 10 years:

\[
V(10) = 25000 \cdot e^{0.14 \cdot 10} = 25000 \cdot e^{1.4}
\]

Let's calculate this:

\[
V(10) \approx 25000 \cdot e^{1.4} \approx 25000 \cdot 4.055 \approx 101375
\]

Now, find the difference between the value after 10 years and 9 years:

\[
\text{Difference} = V(10) - V(9) \approx 101375 - 88100 = 13275
\]

So, the difference in the value is approximately **$13,275**.

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Would you like any more details or explanations on these calculations? Here are some related questions you might find useful:

1. What would be the value of the painting after 5 years?
2. How long would it take for the painting's value to double?
3. What is the percentage increase in the painting's value from year 9 to year 10?
4. If the initial value of the painting was $30,000, how would that affect the value after 9 years?
5. How does the rate $ 0.14 $ affect the growth of the painting's value?

**Tip:** When dealing with exponential growth, the rate of growth significantly impacts the final value over time, especially in longer periods.

Explore how to calculate the value of a painting over time using the exponential growth formula. Solve specific scenarios such as determining initial value, future value after certain years, and differences between values. Gain insights into exponential functions and compound interest applications. Suitable for high school students and enthusiasts of mathematical modeling.

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