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The given equation for the value $ V $ of a painting after $ t $ years from 2007 is:

\[
V = 400000(1.5)^{0.2t}
\]

You are asked to find the value of the painting at the end of 2012. Since the time starts at the beginning of 2007, the value of $ t $ in 2012 would be:

\[
t = 2012 - 2007 = 5 \text{ years}
\]

Now, substitute $ t = 5 $ into the equation:

\[
V = 400000(1.5)^{0.2 \times 5}
\]

This simplifies to:

\[
V = 400000(1.5)^1
\]

\[
V = 400000 \times 1.5 = 600000
\]

Therefore, the value of the painting at the end of 2012 is **$600,000**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the value change if the annual growth rate is altered?
2. What would the value of the painting be in 2020 using the same formula?
3. How would the value change if the initial value was $500,000 instead of $400,000?
4. How does compounding work in similar investment or asset growth scenarios?
5. What if the growth factor was less than 1 (i.e., the painting depreciates)?

**Tip:** Always ensure the units of time align correctly with the exponent when using growth formulas.

The value $V of a famous painting t years after the beginning of 2007 is given by V = 400000(1.5)^{0.2t}. Find the value of the painting at the end of 2012.

Learn how to calculate the value of a famous painting using the exponential growth formula. In this example, find the value of the painting at the end of 2012 given that the value grows based on the equation V = 400000(1.5)^{0.2t}. Suitable for students in Grades 9-12.

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