In​ 2004, an art collector paid

​$151 comma 047 comma 000151,047,000

for a particular painting. The same painting sold for

​$35 comma 00035,000

in 1950. Complete parts​ (a) through​ (d).

Question content area bottom

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.

​V(t)equals=enter your response here

​(Type an expression. Type integers or decimals for any numbers in the expression. Round to three decimal places as​ needed.)

The exponential growth model is given by the formula:

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

Where:
- $ V(t) $ is the value of the painting at time $ t $ (in years after 1950),
- $ V_0 $ is the initial value of the painting (in this case, \$35,000 in 1950),
- $ k $ is the exponential growth rate,
- $ t $ is the number of years after 1950,
- $ e $ is Euler's number (approximately 2.71828).

We know that the painting's value increased from \$35,000 in 1950 to \$151,047,000 in 2004. The time elapsed is:

\[
t = 2004 - 1950 = 54 \text{ years}
\]

### Step 1: Set up the equation using the given values:
\[
151,047,000 = 35,000 \cdot e^{54k}
\]

### Step 2: Solve for $ k $:

1. Divide both sides by 35,000:
\[
\frac{151,047,000}{35,000} = e^{54k}
\]

2. Simplify the left-hand side:
\[
4315.629 = e^{54k}
\]

3. Take the natural logarithm of both sides:
\[
\ln(4315.629) = 54k
\]

4. Calculate $ \ln(4315.629) $:
\[
\ln(4315.629) \approx 8.368
\]

5. Solve for $ k $:
\[
k = \frac{8.368}{54} \approx 0.155
\]

### Step 3: Write the exponential growth function:

\[
V(t) = 35,000 e^{0.155t}
\]

So, the exponential growth function is:

\[
V(t) = 35,000 e^{0.155t}
\]

Would you like further details on how we calculated the growth rate, or any clarification?

Here are some related questions:

1. What is the value of the painting in 1960 using this model?
2. How does the exponential growth model compare to linear growth over 54 years?
3. What would be the painting’s value in 2050 according to this model?
4. Can you derive the formula for $ V(t) $ if the value had decreased instead of grown?
5. How would the growth rate change if the value in 2004 had been different?

**Tip**: When solving exponential growth problems, always ensure you’re using natural logarithms for solving exponential equations involving $ e^x $.

In 2004, an art collector paid $151,047,000 for a particular painting. The same painting sold for $35,000 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.

Calculate the exponential growth rate and find the growth function for a painting's value, increasing from $35,000 in 1950 to $151,047,000 in 2004. This problem uses the exponential growth formula and natural logarithms to determine the rate of growth over 54 years. Suitable for students in Grades 11-12.

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