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.