|
Question 1080651: To print posters for school, the school had initially brought a printing machine (purchased for $300), and they estimated that it will cost them an additional $0.15 to print each poster.
a) Write a linear function C(x) to represent the total cost producing x posters. Include start-up costs.
I think an equation would be C(x) = 300+0.15x
b) Write a rational function A(x) to represent the average cost per poster when x posters are produced.
How would you write a rational function for this?
c) What is the domain of A(x) in the context of this question? Assume that they will not purchase the printing machine without producing any posters.
d) State the horizontal asymptote of A(x) and what does it represent in this question.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Part A) You have the correct cost function. Nice work.
----------------------------------------------------------------------------
Part B)
The average is found by adding up a bunch of values and dividing by the total number of values added up.
For example, the average of the list {1,4,11,15} is (1+4+11+15)/4 = 31/4 = 7.75
In this case, we'll divide the total cost C(x) by the number of items x.
So the average cost function would be
----------------------------------------------------------------------------
Part C)
The domain is the set of allowed inputs for x. The x refers to the number of posters made with this machine. We can't make a negative number of posters, which is why x can't be negative. We also can't make x = 0 copies either because A(x) would be undefined (see part B above; notice the x in the denominator).
So the domain would be  where x is any whole number or integer.
----------------------------------------------------------------------------
Part D)
What happens as x heads off to infinity? Let's first explore a table of values
| x | A(x) |
|---|
| 1 | 300.15 | | 10 | 30.15 | | 100 | 3.15 | | 1,000 | 0.45 | | 10,000 | 0.18 | | 100,000 | 0.153 | | 1,000,000 | 0.1503 | | 10,000,000 | 0.15003 |
Note: each A(x) value is the result of plugging in a given x value which is to the left of it. For example, if x = 10, then plugging this into the A(x) function yields A(10) = 30.15; which means that if you make 10 copies then the average cost per copy is $30.15
As the table suggests, we're approaching an asymptote for large values of x. Specifically we're approaching the asymptote y = 0.15 as x gets larger.
The average cost will never actually get to 15 cents but it will get very close. Though to be fair, with rounding we can effectively get to 15 cents.
The graph below shows the asymptotic behavior
Note: Image generated by GeoGebra (free graphing software).
Which is to be expected. The more copies we make, the more the average cost per copy is to tend toward the actual cost per copy (ignoring fixed cost), which is 15 cents. At first the average cost is much more than 15 cents due to the fixed cost ($300) factored into the cost equation.
|
|
|
| |
