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The given function is:

\[
f(x) = 1.2x^3 - 32x^2 + 261x + 562
\]

This models the number of discharges from the military, \( f(x) \), of active-duty gay service members under the "don't ask, don't tell" policy, \( x \) years after 1994.

### Part a: Find the slope of the secant line from \( x_1 = 0 \) to \( x_2 = 4 \)

The slope of a secant line between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is calculated using the slope formula:

\[
\text{slope of secant line} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

#### Step 1: Calculate \( f(0) \) and \( f(4) \)

Using the given function:
- \( f(0) = 1.2(0)^3 - 32(0)^2 + 261(0) + 562 = 562 \)
- \( f(4) = 1.2(4)^3 - 32(4)^2 + 261(4) + 562 \)

Let's calculate \( f(4) \).The values of the function are:
- \( f(0) = 562 \)
- \( f(4) = 1170.8 \)

#### Step 2: Calculate the slope of the secant line

\[
\text{slope} = \frac{f(4) - f(0)}{4 - 0} = \frac{1170.8 - 562}{4} = \frac{608.8}{4} = 152.2
\]

Rounding to the nearest whole number:

\[
\text{slope} \approx 152
\]

Thus, the slope of the secant line is approximately **152**.

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Would you like any further details or have any other questions?

Here are 5 related questions:
1. How would you find the instantaneous rate of change of the function at \( x = 4 \)?
2. What is the interpretation of the secant line's slope in this context?
3. How would the slope change if we used different values for \( x_1 \) and \( x_2 \)?
4. What are the units of the slope in this scenario?
5. How could you determine the point where the function reaches its maximum?

**Tip:** The slope of the secant line gives the average rate of change between two points, which can provide insights into overall trends over time.

The function f(x) = 1.2x^3 - 32x^2 + 261x + 562 models the number of discharges from the military, f(x), of active-duty gay service members under the "don't ask, don't tell" policy x years after 1994. Find the slope of the secant line from x1 = 0 to x2 = 4.

Learn how to calculate the slope of the secant line from x1 = 0 to x2 = 4 for the function f(x) = 1.2x^3 - 32x^2 + 261x + 562. This problem explores concepts in calculus, including secant lines and the average rate of change, suitable for advanced high school or early college students.

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