This question is asking you to find a formula for the cost function $C(x)$, which represents the cost to produce $x$ cups of coffee.

From the image, you are provided with two pieces of information:

1. The cost to produce 200 cups of coffee is $19.93.
2. The cost to produce 400 cups of coffee is $38.53.

Since the problem states that the cost $C(x)$ is a linear function, we can assume the general form of the equation to be: $C(x) = mx + b$ where:

• $m$ is the slope (the rate of change of the cost with respect to the number of cups),
• $b$ is the fixed cost (y-intercept) when no cups are produced.

Step 1: Find the slope ($m$)

The slope $m$ can be calculated using the two given points: $(200, 19.93) \quad \text{and} \quad (400, 38.53)$ Using the formula for the slope: $m = \frac{C(400) - C(200)}{400 - 200} = \frac{38.53 - 19.93}{400 - 200}$ Let's calculate this.The slope $m$, representing the cost increase per additional cup, is approximately 0.093.

Step 2: Find the intercept ($b$)

Now that we know the slope, we can substitute one of the points (e.g., $(200, 19.93)$) into the linear equation $C(x) = mx + b$ to solve for $b$:

$19.93 = 0.093(200) + b$ Let's solve for $b$.The y-intercept $b$, representing the fixed cost, is approximately 1.33.

Final Equation:

Thus, the linear cost function is: $C(x) = 0.093x + 1.33$ This formula gives the total cost to produce $x$ cups of coffee.

Would you like further details or have any other questions?

Here are some related questions:

1. How do we interpret the slope in the context of this problem?
2. What would the cost be to produce 300 cups of coffee?
3. Can a linear