In​ 2004, an art collector paid 

​$66 comma 870 comma 00066,870,000 

for a particular painting. The same painting sold for 

​$33 comma 00033,000 

in 1950.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.

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(t) $ is the value of the painting after $ t $ years,
- $ V_0 $ is the initial value of the painting (in 1950),
- $ k $ is the growth rate we want to find,
- $ t $ is the number of years since 1950.

### Step 1: Plug in the known values
- The value of the painting in 1950 was $ V_0 = 33,000 $ dollars.
- The value of the painting in 2004 was $ V(2004 - 1950) = V(54) = 66,870,000 $ dollars.
- We are given that $ t = 54 $ years.

So the equation becomes:
\[
66,870,000 = 33,000 \cdot e^{54k}
\]

### Step 2: Solve for $ k $
1. Divide both sides of the equation by $ 33,000 $:
\[
\frac{66,870,000}{33,000} = e^{54k}
\]
\[
2020 = e^{54k}
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(2020) = 54k
\]

3. Solve for $ k $:
\[
k = \frac{\ln(2020)}{54}
\]

Using a calculator to evaluate $ \ln(2020) $:
\[
\ln(2020) \approx 7.611
\]

Thus:
\[
k = \frac{7.611}{54} \approx 0.141
\]

So, the exponential growth rate $ k \approx 0.141 $.

### Step 3: Write the exponential growth function $ V(t) $

Now that we have $ k $, we can write the function for $ V(t) $:
\[
V(t) = 33,000 \cdot e^{0.141t}
\]

This is the exponential growth function that models the value of the painting after $ t $ years since 1950.

---

Do you want further details or have any questions?

### Related Questions:
1. How would the value of the painting change if $ k $ were higher?
2. What would the value of the painting be in the year 2025 using this function?
3. How long would it take for the painting's value to double from its 1950 price?
4. What is the significance of the exponential growth rate in this context?
5. How can you interpret $ e $ in terms of exponential growth?

### Tip:
When dealing with exponential growth or decay problems, always check the units of time to ensure consistency in calculations.

In 2004, an art collector paid $66,870,000 for a particular painting. The same painting sold for $33,000 in 1950. 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.

Find out how to calculate the exponential growth rate and growth function for a painting that increased in value from $33,000 in 1950 to $66,870,000 in 2004. Learn about key concepts in exponential growth, including solving for the growth rate using logarithms, suitable for students in Grades 10-12.

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