if the cost to to produce 100 cups of coffee is 11.02, while the cost to produce 400 cups is 401.12, what is the linear cost function?

To find the linear cost function, we need to determine the relationship between the number of cups produced and the cost of production. In this case, we can consider the number of cups produced as the independent variable (x) and the cost of production as the dependent variable (y).

We are given two data points: (100, 11.02) and (400, 401.12). We can use these two points to find the slope (m) and the y-intercept (b) of the linear cost function.

First, let's calculate the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (401.12 - 11.02) / (400 - 100)
m = 390.10 / 300
m = 1.30

The slope represents the change in cost for each additional cup produced. In this case, every additional cup of coffee increases the cost by $1.30.

Next, let's calculate the y-intercept (b). We can use one of the given data points, (100, 11.02), and substitute it into the slope-intercept form of a linear equation (y = mx + b) to solve for b:
11.02 = 1.30 * 100 + b
11.02 = 130 + b
b = 11.02 - 130
b = -118.98

Now we have the slope (m = 1.30) and the y-intercept (b = -118.98), we can write the linear cost function as:

Cost = 1.30x - 118.98

Therefore, the linear cost function is Cost = 1.30x - 118.98, where x represents the number of cups produced.