To find the cost function, we first need to establish a linear model. We begin by finding the slope (m), which is the rate of change. Then, we'll find the y-intercept (b), which is the initial value of y when x is zero.
(a) To find the slope, we will take the difference in cost and divide it by the difference in units. We have two points from the question (100, $10,000) and (150, $12,000). The slope is calculated as follows:
m = (Y2 - Y1) / (X2 - X1)
Here, X1 is 100 units, Y1 is $10,000, X2 is 150 units, and Y2 is $12,000.
So, we calculate the slope, m, as:
m = ($12,000 - $10,000) / (150 - 100)
= $2,000 / 50
= $40.
Thus, each additional unit increases the cost by $40.
Then, we calculate the y-intercept. The general equation for a linear model is Y = mX + b, where b is the y-intercept.
We can use one of the given points and the slope to find b. Let's use the first point (100, $10,000).
Plugging the slope and the point values into the equation gives:
$10,000 = $40 * 100 + b
Solving for b gives:
b = $10,000 - $4,000 = $6,000.
So, the cost function for the linear model, based on the given points, is:
Cost = $40 * Quantity + $6,000.
This means that the fixed cost is $6,000 and each additional AC unit costs an extra $40 to produce.