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.