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The image contains a problem related to the total worldwide box-office receipts for a long-running blockbuster movie, modeled by the function:

\[
T(x) = 250 \left(1 - e^{-0.3x}\right)
\]

where $ T(x) $ is measured in millions of dollars, and $ x $ is the number of months since the movie's release. The problem asks several questions, including:

1. The total box-office receipts after the first month, second month, third month, and eventually as $ x \to \infty $ (the long-term total).

Let's break down and solve the problem step by step.

### 1. Total box-office receipts after the first month ($ x = 1 $):
\[
T(1) = 250 \left(1 - e^{-0.3(1)}\right) = 250 \left(1 - e^{-0.3}\right)
\]
Calculating this value numerically.

### 2. Total box-office receipts after the second month ($ x = 2 $):
\[
T(2) = 250 \left(1 - e^{-0.3(2)}\right) = 250 \left(1 - e^{-0.6}\right)
\]
We'll compute this as well.

### 3. Total box-office receipts after the third month ($ x = 3 $):
\[
T(3) = 250 \left(1 - e^{-0.3(3)}\right) = 250 \left(1 - e^{-0.9}\right)
\]
Finally, for the long-term receipts:

### 4. Total box-office receipts as $ x \to \infty $:
\[
T(\infty) = 250 \left(1 - e^{-0.3 \cdot \infty}\right) = 250 \times 1 = 250 \, \text{million dollars}
\]
This represents the asymptotic total.

I'll now calculate the specific numerical values for the first three months using Python.The total box-office receipts for the movie, based on the function, are as follows:

1. **After the first month**: \$64.80 million
2. **After the second month**: \$112.80 million
3. **After the third month**: \$148.36 million
4. **Long-term total (as $ x \to \infty $)**: \$250 million

These values match with the behavior of the exponential model, where the movie's total earnings gradually approach \$250 million over time.

Would you like further clarification or details on any specific aspect? Here are some related questions for further exploration:

1. How does the exponential function model gradual growth?
2. Can we find the rate at which the movie's earnings are increasing in the first few months?
3. What would happen if the growth rate ($0.3$) were different?
4. How long does it take for the earnings to reach 90% of the total?
5. How does the function behave when different initial total receipts are used?

**Tip**: Exponential models are often used in real-world applications, like finance and population growth, due to their rapid initial increase followed by gradual leveling off.

The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function T(x) = 250(1 - e^(-0.3x)) where T(x) is measured in millions of dollars, and x is the number of months since the movie’s release. What are the total box-office receipts after the first month, second month, third month? What will the movie gross in the long run (as x approaches infinity)?

Explore how to model box-office receipts using the exponential function T(x) = 250(1 - e^(-0.3x)). This problem walks through finding total receipts after the first, second, and third months, and the eventual long-term gross. Suitable for Grades 10-12.

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