To find the formula for $C(x)$, where $C(x)$ represents the cost to produce $x$ cups of coffee and it's a linear function, we can model this using the slope-intercept form of a linear equation:

$C(x) = mx + b$

Here:

• $m$ is the slope of the line (the rate of change of cost per cup),
• $b$ is the y-intercept (the fixed cost when no cups are produced),
• $x$ is the number of cups produced.

Step 1: Determine the slope $m$

We are given two points: $(200, 19.92)$ and $(600, 57.12)$, where the first number represents the number of cups and the second represents the cost. The slope $m$ is calculated as:

$m = \frac{C(600) - C(200)}{600 - 200}$

Substitute the given values:

$m = \frac{57.12 - 19.92}{600 - 200} = \frac{37.2}{400} = 0.093$

So, the slope $m = 0.093$. This means the cost increases by 9.3 cents for each additional cup of coffee.

Step 2: Find the y-intercept $b$

To find $b$, substitute one of the points into the equation $C(x) = 0.093x + b$. We'll use the point $(200, 19.92)$:

$19.92 = 0.093(200) + b$

Simplifying:

$19.92 = 18.6 + b$

$b = 19.92 - 18.6 = 1.32$

Step 3: Write the formula for $C(x)$

Now that we have both the slope $m = 0.093$ and the y-intercept $b = 1.32$, the formula for the cost function is:

$C(x) = 0.093x + 1.32$

This is the linear function that represents the cost to produce $x$ cups of coffee.

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Would you like a breakdown of any specific step?

Here are five related questions to further your understanding:

1. How do you interpret the slope $m$ in the context of this problem?
2. What does the y-intercept $b$ represent in a real-world situation?
3. How would the formula change if producing 600 cups cost less than expected, say $50?
4. Can you use this formula to predict the cost of producing 1,000 cups of coffee?
5. How would the equation be different if the cost was quadratic rather than linear?

Tip: When modeling with linear functions, always check the units of the slope to ensure you interpret the rate of change correctly.