If we produce 350 units of output, the total cost is $11,500. It costs us $12,900 to produce 420 units. If we assume the relationship is linear, determine the cost function.
treat your data as two ordered pairs
( 350 , 11500) and (420 , 12900)
find the equation like you would given any two ordered pairs,
i.e.
find the slope etc
cost = slope (number of units) + constant
y = m x + b
11500 = 350 m + b
12900 = 420 m + b
solve for m and b by elimination or substitution
To determine the cost function, we need to find the equation that relates the number of units produced to the total cost. Given that the relationship is assumed to be linear, we can use the formula for the equation of a line.
Let's denote the number of units produced as x and the total cost as y. We have two data points: (350, $11,500) and (420, $12,900).
First, we need to find the slope of the line. The slope, denoted as m, represents the change in y divided by the change in x. We can calculate it using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values from our data points:
m = (12,900 - 11,500) / (420 - 350)
m = 1,400 / 70
m = 20
Now that we have the slope, we can find the y-intercept, denoted as b. The y-intercept represents the value of y (total cost) when x (number of units produced) is zero. To find the y-intercept, we can use the formula:
b = y - mx
Using one of our data points, (350, $11,500):
11,500 = 350(20) + b
b = 11,500 - 7,000
b = 4,500
Thus, the equation for the cost function is:
y = 20x + 4,500
The cost function represents the relationship between the number of units produced (x) and the total cost (y).