Course:MATH110/Archive/2010-2011/003/Teams/Schwytz/Homework 11
Write a linear model to predict the cost of producing flags of your teams Canton under the assumptions that the marginal cost is $7 per unit and that at the current production level of 20 items the cost is $100.
Our model is described by a linear function.
- We know that our model has to pass through the point (20,100) because the question tells us that the current production level of 20 items costs $100 (where the cost in dollar is the y-axis and the number of items is the x-axis) therefore we can use this point to formulate a function using the standard equation formula (y=mx +b)
- We can use 20 items for ‘x’ and 100 dollars for ‘y’
- Now we must find ‘m’
- To do this we need to use the formula: y2-y1/ x2-x1
To find a point on the graph:
- We know that 20 items cost 100 dollars and the marginal cost is $7 per unit
- Therefore we know that 21 items costs $107 and then 22 items costs $114
- Now we can set this up in the above equation (where the cost is ‘y’ and the items is ‘x’:
114-107/ 22-21 = 7
- Therefore:
m=7
- Now we need to find 'b':
y=mx + b
107 = 7 (21) +b
b = -40
- Therefore the equation of the line is:
y= 7x - 40 , which is a linear model
- If we were to graph this model the function would cross the y-axis at -40.
- However we can also determine the value of ‘b’ by understanding that it costs $100 to produce 20 flags which averages to $5 per flag. If the flags were $7 each it would have cost $140. The difference between these two costs is $40. Therefore to take into account the saving of $40 we make ‘b’ -40 in the formula y=mx+b
- To relate this formula to the specific terminology of the question we will replace 'x' with 'f' and let 'f' represent the number of flags produced (where 'f' must be greater than 20). We will replace 'y' with C(f) and let C(f) represent the cost to produce the flags:
C(f) = 7f - 40 , where f ≥ 20
What does your model predict for production of a 150 items?
- In order to predict the total cost of the production of 150 items we can substitute into our model 150 for 'f' . Which gives us the equation:
C(f)=7(150)-40
C(f)=1010
According to your model, what happens when the average cost per item, as production levels increase?
- In order to determine the average cost per item as production increases we must first find the total cost of three consecutive units (e.g the total cost of producing 23, 24, and 25 flags) and divide each total you get but the number of flags produced as shown below:
The average cost equation: total cost/number of flags
1. Let f=23
C(f)=7(23)-40
C(f)=121
Average cost of 23 units is : 121/23=5.26
2. Let f=24
C(f)=7(24)-40
C(f)=128
Average cost of 24 units is : 128/24 =5.33
3. Let f=25
C(f)=7(25)-40
C(f)=135
Average cost of 25 units is: 135/25=5.40
- Based on the information above we can see that the averge cost per item increases as the production levels increase.
Finally, find some other models ( not necessarily linear) for which you get other behaviours such as
1. The average cost remains constant as production increases:
Model: y=3x
- This model increases at a constant rate, because the slope of the function is the same everywhere. This model shows that the average cost will therefore remain constant as the production increases.
2. The average cost diminishes as the production increases:
Model: y= x^1/2
- For this model when you plug in any postive numbers in for 'x', you will see that as 'x' or the number of items produced increases, the average cost decreases:
ex.
y= x^1/2 , let x = 100 and let y represent the total cost of x
y= (100^1/2)/ 100
y= 0.1
Now let x= 1000000
y= (1000000^1/2)/1000000
y= 0.001
- Here you can see that the average cost is diminishing as the production ('x') increases.
3. The average cost increases as production increases:
Model: y= e^x
- In this model as the production increases so does the slope of the model which causes the average cost to also therefore increase.
4. You obtain an economy of scale. This means that staring at some specific production level, the marginal cost is always less than the average cost:
Model: y= x^1/2
- For this model, we can see that the marginal cost is less than the average cost
ex.
let x=3
y=3^1/2
y= 1.73
- now we must divide by 3 to get the average
y= 1.73/3
y= 0.577 , this means that it costs an average of $0.58 to produce 3 items
- And when x=2, y= 1.41
- Therefore if we calculate the difference ($1.73-$1.41) we get 0.318 or $0.32. This means that it only cost $0.32 to produce the third item but it cost an average of $0.58 per item (of the 3 total items). This model therefore obtains an economy of scale.