The manager of a restaurant found that the cost to produce 200 cups of coffee is ​$19.92​, while the cost to produce 600 cups is ​$57.12. Assume the cost​ C(x) is a linear function of​ x, the number of cups produced.Find a formula for C(x)

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.

The manager of a restaurant found that the cost to produce 200 cups of coffee is ​$19.92​, while the cost to produce 600 cups is ​$57.12. Assume the cost​ C(x) is a linear function of​ x, the number of cups produced. Find a formula for C(x).

Learn how to find the linear cost function C(x) to produce cups of coffee, given that 200 cups cost $19.92 and 600 cups cost $57.12. This problem covers key algebraic concepts, including slope-intercept form and linear equations. Suitable for students in Grades 9-11.

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