In​ 2004, an art collector paid ​$77 comma 963 comma 000 for a particular painting. The same painting sold for ​$31 comma 000 in 1950. Complete parts​ (a) through​ (d).
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Part 1
​a) Find the exponential growth rate​ k, to three decimal​ places, and determine the exponential growth function​ V, for which​ V(t) is the​ painting's value, in​ dollars, t years after 1950.
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​(Do not round until the final answer. Then round to three decimal places as​ needed.)

To find the exponential growth rate $ k $ and the exponential growth function $ V(t) $, we can use the formula for exponential growth:

\[
V(t) = V_0 e^{kt}
\]

where:
- $ V_0 $ is the initial value of the painting,
- $ k $ is the growth rate,
- $ t $ is the time in years since 1950.

### Given Data
- Initial value $ V_0 = 31,000 $ (in 1950)
- Value in 2004 $ V(54) = 77,963,000 $ (since $ t = 2004 - 1950 = 54 $)

### Step 1: Set up the equation
We know that:

\[
V(54) = V_0 e^{k \cdot 54}
\]

Substituting the values:

\[
77,963,000 = 31,000 e^{54k}
\]

### Step 2: Solve for $ k $
First, isolate the exponential term:

\[
e^{54k} = \frac{77,963,000}{31,000}
\]

Calculating the right side:

\[
e^{54k} \approx 2510.16129
\]

Now, take the natural logarithm of both sides:

\[
54k = \ln(2510.16129)
\]

Calculating the natural logarithm:

\[
\ln(2510.16129) \approx 7.825
\]

Now, solve for $ k $:

\[
k = \frac{7.825}{54} \approx 0.145
\]

### Step 3: Round $ k $
To three decimal places, we have:

\[
k \approx 0.145
\]

### Step 4: Write the exponential growth function
Now we can write the exponential growth function:

\[
V(t) = 31,000 e^{0.145t}
\]

### Summary
- $ k \approx 0.145 $
- The exponential growth function is $ V(t) = 31,000 e^{0.145t} $

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How does the choice of initial value impact the growth rate?
2. What would be the painting's value in another year, such as 1960?
3. How can we determine the time it takes for the painting to double in value?
4. What factors might influence the growth rate of art values over time?
5. How does this growth rate compare to other investments, like stocks or real estate?

**Tip:** Always check your calculations for accuracy, especially when working with logarithms and exponential functions!

In 2004, an art collector paid $77,963,000 for a particular painting. The same painting sold for $31,000 in 1950. Complete parts (a) through (d). a) Find the exponential growth rate k, to three decimal places, and determine the exponential growth function V, for which V(t) is the painting's value, in dollars, t years after 1950.

Explore how to find the exponential growth rate k and the function V(t) for a painting's value, which appreciated from $31,000 in 1950 to $77,963,000 in 2004. This example illustrates key concepts in exponential growth suitable for advanced high school students.

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