A product may be made using Machine I or Machine II. The manufacturer estimates that the monthly fixed costs of using Machine I are $17,000, whereas the monthly fixed costs of using Machine II are $14,000. The variable costs of manufacturing 1 unit of the product using Machine I and Machine II are $13 and $19, respectively. The product sells for $60 each.

Question

A product may be made using Machine I or Machine II. The manufacturer estimates that the monthly fixed costs of using Machine I are $17,000, whereas the monthly fixed costs of using Machine II are $14,000. The variable costs of manufacturing 1 unit of the product using Machine I and Machine II are $13 and $19, respectively. The product sells for $60 each.

(a) Find the cost functions C(x) associated with using each machine, where x is the number of units produced using that machine.

Machine I C1(x) = 17000 + 13x

Machine II C2(x) = 14000 + 19x

Graph the following equations,

Answer 1

y = 17000 + 13x

y = 14000 + 19x

Answer 2

To graph the cost functions C1(x) and C2(x), we need to plot the cost as a function of the number of units produced (x) for each machine.

The cost function C1(x) for Machine I is given by: C1(x) = 17000 + 13x
The fixed cost is $17,000, and the variable cost per unit is $13.

Similarly, the cost function C2(x) for Machine II is given by: C2(x) = 14000 + 19x
The fixed cost is $14,000, and the variable cost per unit is $19.

Now, let's plot the graphs:

For C1(x), the y-intercept represents the fixed cost ($17,000), and the slope represents the variable cost per unit ($13). Since the slope is positive, the graph will have an upward slope.

For C2(x), the y-intercept represents the fixed cost ($14,000), and the slope represents the variable cost per unit ($19). Again, the slope is positive, indicating an upward slope.

We can choose a range of values for x and calculate the corresponding costs for both machines to plot the graphs accurately. Let's say we choose x to be between 0 and 100 units.

Now, let's plot the graphs of C1(x) and C2(x) on a coordinate plane with x-axis representing the number of units produced and y-axis representing the cost:

- Graph of C1(x): Plot the points (0, 17000) and (100, 17000 + 13*100) and draw a line passing through the two points.

- Graph of C2(x): Plot the points (0, 14000) and (100, 14000 + 19*100) and draw a line passing through the two points.

This will give you the graphical representation of the cost functions C1(x) and C2(x) for Machine I and Machine II, respectively.

Answer 3

To graph the cost functions C1(x) and C2(x), we can plot some points for each function and then connect them to form the lines.

For Machine I:
Let's choose two values of x, say x=0 and x=100, to calculate the corresponding costs:
- For x = 0, C1(x) = 17000 + 13(0) = 17000.
- For x = 100, C1(x) = 17000 + 13(100) = 17000 + 1300 = 18300.

So we have two points: (0, 17000) and (100, 18300).

For Machine II:
Using the same approach, let's calculate the costs for x = 0 and x = 100:
- For x = 0, C2(x) = 14000 + 19(0) = 14000.
- For x = 100, C2(x) = 14000 + 19(100) = 14000 + 1900 = 15900.

So we have two points: (0, 14000) and (100, 15900).

Now we can plot these points and draw the lines:

For Machine I:
- The point (0, 17000) represents the fixed cost.
- The point (100, 18300) represents the total cost when producing 100 units.

For Machine II:
- The point (0, 14000) represents the fixed cost.
- The point (100, 15900) represents the total cost when producing 100 units.

(Note that the y-axis represents the cost, and the x-axis represents the number of units produced.)

Graphically, the lines representing the cost functions C1(x) and C2(x) will have a positive slope, as the costs increase with the number of units produced. The line for Machine I will start at (0, 17000) and rise less steeply than the line for Machine II, which will start at (0, 14000) and rise more steeply.

I hope this helps!