MHF4U
U02A08 Homework Solutions
Assignment Solutions
1. An ice skate manufacturer has its cost function defined as: C(x) = 2500 + 20x where C(x) is the cost of
producing x pairs of skates.
a) Write an average cost function A(x) for producing x pairs of skates.
Solution: Average cost will depend on total cost being divided by the number of items being produced.
C ( x)
x
2500  20 x
A( x) 
x
A( x) 
b) What is the domain of A(x)? Explain the elements of the domain in the context of the problem.
Solution: For the domain, we look to the denominator and set it equal to zero. This will give us the value
of x that would give an indeterminate form. {x  R | x  0}
Since we are looking at the number of pairs of skates, x will be need to be a positive value. Thus the
domain is: {x  R | x  0}
c) What is the horizontal asymptote for this function? What does it represent in the context of the problem?
Is this situation realistic?
Solution: the horizontal asymptote is y = 20. This represents the cost per pair of skates as the number of
pairs of skates produced increases without bound. The cost of production will never get to $20 per pair,
however, as there are limits to the production capabilities of the factory and the staff. Also, as time passes,
the costs per pair will change, so this is likely just a short term model for the average cost function.
Salaries and materials will increase in cost over time, and thus the model will need to be adjusted.
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MHF4U
U02A08 Homework Solutions
2. Mark and Heather are graphing rational functions for their class.
Mark is graphing the function: f ( x )  3 x  5
x
Heather is graphing the function g ( x )  3  5
x
The teacher asks each of them the following questions:
a) What is the domain of your function?
Mark: : { x  R |
x  0}
Heather: : { x  R |
x  0}
b) What are the asymptotes of your function?
Mark:
Vertical asymptote:
x0
Horizontal asymptote: y = 3.
Heather:
Vertical asymptote: x = 0
Horizontal asymptote: y = 3
c) Sketch each of the functions. Compare each of the function's graphs.
What can you tell me about these two functions?
Mark
Heather
10
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
2
-2
-4
-4
-6
-6
-8
-8
-10
-10
Mark and Heather have identical functions.
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-10 -8 -6 -4 -2
-2
4
6
8 10
MHF4U
U02A08 Homework Solutions
d) Explain what information would have indicated that these functions had strong commonalities.



Same asymptotes
Same domain
Same coefficients for the terms in the function – note that if Heather were to get a common
denominator in her function, it would be exactly the same as Mark’s!
3. A home computer costs $9/year in electricity costs. Assume a new computer costs $2000 to purchase.
a) Determine a function that gives the annual cost of a computer as a function of the number of years you
own the computer.
A( x ) 




2000  9 x
x
A(x) represents the average annual cost
2000 would be the initial cost of buying the computer
9 would represent the cost of electricity per year
x would represent the number of years the computer is owned and operated
b) Sketch the graph of the function you have developed in (b).
3000
2700
Average Annual
Cost of the
computer
2400
2100
1800
1500
1200
900
600
300
1
Years
2
3
4
5
6
7
8
9 10
c) What are the asymptotes for this function and what do they mean in the context of the problem? Is this
realistic?
The vertical asymptote is : x = 0. To discuss and calculate annual costs of owning a computer, you have
at least buy one computer.
Horizontal Asymptote(s): y = 9
As the years go on the annual cost will tend to $9. As the years increase, the set-up costs spread out over
a large number of years. Remember the annual cost will never actually reach $9 but it will get closer and
closer to $9 as time goes by.
We are assuming that the life of the computer is eternal. We know that this not true and that eventually
machinery will break down and/or become obsolete.
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MHF4U
U02A08 Homework Solutions
4. The concentration, c (micrograms per millilitre (μg/mL) ) of a drug in the blood, t hours after the drug is
taken orally, is given by:
c(t) 
5t
.
2
2t  7
a) What is the domain of the function, in the context of this question?
The domain of the function, based strictly on the algebraic expression, is: {t  }
However, in the context of this question, since t represents time, the domain must be restricted to:
{t   | t  0}
b) Find any vertical and horizontal asymptotes for this function. Interpret their meaning in the context of the
question.
There are no vertical asymptotes, since there are no restrictions on the domain of this function in the
algebraic sense.
The horizontal asymptote is
c  0.
This is discovered by looking at the terms with the highest degree of t in each of the numerator and
denominator:
5t
c 2
2t
5
c
2t
As
t  ; c  0
This suggests that as time passes, the concentration of the drug is reducing to zero, though it will
never actually attain that level.
c) Graph the function, using the Trace feature. What is the maximum concentration of drug in the blood?
How long does it take to reach this maximum?
Maximum concentration occurs at about time = 1.60 hours, and is 0.66 μg/mL.
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