Assume it costs Microsoft $4,600 to manufacture 5 Xbox 360s and $7,200 to manufacture 10. Obtain the corresponding linear cost function.
C(x) =

What was the cost (in dollars) to manufacture each additional Xbox?
$
Use the cost function to estimate the cost of manufacturing 55 Xboxes (in dollars).
$

We are given two points that correspond to the number of Xbox 360 units and their associated costs:

1. $ (5, 4600) $ — it costs $4600 to manufacture 5 Xboxes.
2. $ (10, 7200) $ — it costs $7200 to manufacture 10 Xboxes.

To obtain the linear cost function $ C(x) = mx + b $, we first need to determine the slope $ m $, which represents the cost of manufacturing each additional Xbox.

### Step 1: Find the slope $ m $
The slope of a line is given by:

\[
m = \frac{C_2 - C_1}{x_2 - x_1}
\]

Where:
- $ C_1 = 4600 $, $ x_1 = 5 $
- $ C_2 = 7200 $, $ x_2 = 10 $

\[
m = \frac{7200 - 4600}{10 - 5} = \frac{2600}{5} = 520
\]

So, the cost to manufacture each additional Xbox is **$520**.

### Step 2: Find the y-intercept $ b $
Now that we know $ m = 520 $, we can use one of the points to find $ b $ by substituting into the linear equation $ C(x) = mx + b $. Using the point $ (5, 4600) $:

\[
4600 = 520(5) + b
\]

\[
4600 = 2600 + b
\]

\[
b = 4600 - 2600 = 2000
\]

Thus, the linear cost function is:

\[
C(x) = 520x + 2000
\]

### Step 3: Estimate the cost of manufacturing 55 Xboxes
Using the cost function $ C(x) = 520x + 2000 $, we can find the cost of manufacturing 55 Xboxes by substituting $ x = 55 $:

\[
C(55) = 520(55) + 2000 = 28600 + 2000 = 30600
\]

Thus, the estimated cost to manufacture 55 Xboxes is **$30,600**.

---

### Summary:
1. **Cost function**: $ C(x) = 520x + 2000 $
2. **Cost to manufacture each additional Xbox**: **$520**
3. **Cost to manufacture 55 Xboxes**: **$30,600**

Would you like more details on any step?

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Here are 5 related questions:
1. How would the cost function change if the cost per additional Xbox decreased over time?
2. How would you model costs if there were a fixed manufacturing setup fee?
3. How do you find the break-even point in such cost functions?
4. Can this linear model be used for larger scales of production (e.g., 1000 Xboxes)?
5. What are the limitations of using a linear cost function in manufacturing?

**Tip**: The slope in a cost function represents the marginal cost, which is key in business for understanding how much it costs to produce one more unit of a product.

Assume it costs Microsoft $4,600 to manufacture 5 Xbox 360s and $7,200 to manufacture 10. Obtain the corresponding linear cost function. What was the cost (in dollars) to manufacture each additional Xbox? Use the cost function to estimate the cost of manufacturing 55 Xboxes.

Learn how to calculate the linear cost function based on the manufacturing costs for Xboxes. The problem involves determining the cost of producing each additional unit and estimating the cost for manufacturing 55 units using the slope-intercept form of a linear equation. Suitable for students in Grades 9-11.

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