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.

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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.