If it costs Microsoft $4,110 to manufacture 8 XBox Series X consoles in an hour and $8,190 to manufacture 16 in an hour at a particular plant, obtain the corresponding linear cost function, with C(x) in dollars.†
C(x) =

What was the cost (in dollars) to manufacture each additional console?
$
Use the cost function to estimate the cost (in dollars) of manufacturing 35 consoles in an hour.
$

We are given two points corresponding to the number of consoles and their respective cost. These points are:

- $ (x_1, C_1) = (8, 4110) $
- $ (x_2, C_2) = (16, 8190) $

### Step 1: Find the slope of the linear function
The cost function $ C(x) $ is linear, so it has the form:

\[
C(x) = mx + b
\]

where $ m $ is the slope and $ b $ is the y-intercept. The slope $ m $ can be calculated using the formula:

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

Substituting the given values:

\[
m = \frac{8190 - 4110}{16 - 8} = \frac{4080}{8} = 510
\]

So, the slope is $ m = 510 $. This means that the cost to manufacture each additional console is **$510**.

### Step 2: Use the slope to find the y-intercept
We now substitute one of the points (for example, $ x = 8 $ and $ C(x) = 4110 $) into the equation $ C(x) = 510x + b $ to find the y-intercept $ b $:

\[
4110 = 510(8) + b
\]
\[
4110 = 4080 + b
\]
\[
b = 4110 - 4080 = 30
\]

So, the y-intercept is $ b = 30 $.

### Step 3: Write the cost function
Now that we have both the slope and the y-intercept, the cost function is:

\[
C(x) = 510x + 30
\]

### Step 4: Estimate the cost of manufacturing 35 consoles
To estimate the cost of manufacturing 35 consoles, substitute $ x = 35 $ into the cost function:

\[
C(35) = 510(35) + 30 = 17850 + 30 = 17880
\]

So, the estimated cost to manufacture 35 consoles is **$17,880**.

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Do you want more details or have any questions?

Here are some related questions:
1. How would the cost function change if the cost for 16 consoles were $9,000?
2. What is the break-even point if the selling price per console is $600?
3. How would you interpret the y-intercept of this cost function?
4. What would the cost function be if the price per additional console decreased by $20?
5. How does changing the slope affect the overall manufacturing cost?

**Tip:** In linear cost functions, the slope represents the variable cost per unit, while the y-intercept represents the fixed cost of production.

If it costs Microsoft $4,110 to manufacture 8 XBox Series X consoles in an hour and $8,190 to manufacture 16 in an hour at a particular plant, obtain the corresponding linear cost function, with C(x) in dollars. What was the cost (in dollars) to manufacture each additional console? Use the cost function to estimate the cost (in dollars) of manufacturing 35 consoles in an hour.

Learn how to calculate the linear cost function for manufacturing XBox Series X consoles, given the costs for 8 and 16 units. Discover how to find the cost per additional console and estimate the total cost for manufacturing 35 consoles using algebraic concepts such as slope and linear functions.

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