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The graph above shows the total costs for
copying jobs at three different copying services, R, S,
and T. Based on the information represented in the graph,
answer each of the questions below.
a. Which service (R, S, or T) charges the least to make
200 copies?
b. What are the least number and the greatest number of
copies for which copying service S has the lowest cost?
c. If the number of copies for a certain job is increased
by 100, which service (R, S, or T) charges the most for
these extra copies? Give a reason for your answer.
SOLUTIONS AND DISCUSSION QUESTIONS
We recommend that when you review the solutions to the
Copy Costs task with your students, you also ask
questions that encourage a deep understanding of the mathematics
related to this problem. There are suggested questions below
that encourage deep understanding. You should consider using
them as part of the discussion of the solutions to the task.
The answers given are not necessarily the only acceptable
answers since answers may vary depending on the question
asked.
Part a.
Service R
Ask students:
- How do you know that service R costs the least to
make 200 copies?
Answer: When
the number of copies is 200 (along horizontal-axis), line
R is the lowest line (vertical values indicate total cost).
- Does service R cost the least to make any number
of copies?
Answer: No,
because after 500 copies, line R is never the lowest.
- For what numbers of copies does service R cost the
least?
Answer: It
costs the least when the number of copies is between 0
and 500 (for 500 copies, R & S have equal costs).
Part b. Least
501, the greatest 1499 (500 and 1500 are also acceptable)
Ask students:
- How do you know that service S has the lowest total
cost over this interval?
Answer: Line
S is lower than the other lines front 500 copies to 1500
copies (for 500 copies, R & S have equal costs and
for 1500 copies, T & S have the same).
- Which service has the lowest total cost when the
number of copies is greater than 1500? Why?
Answer: Line
T because it is the lowest line after the number of copies
exceeds 1500.
- Which service has the lowest total cost when the
number of copies is less than 500?
Answer: Line
R because it is the lowest line when the number of copies
is less than 500.
Part c.
R because it has the greatest slope or
R because the line is the steepest or
R because it has the greatest rate or
R because it has the greatest cost per copy
Ask students:
- How is the steepness of the lines related to the
extra cost per copy at each service?
Answer: The steeper
the line, the greater the increase in cost for a given
increase in copies. That is, for a given run, the line
with the greatest rise represents the greatest increase
in cost.
- To calculate the slopes of the lines and express
them as rates of change with a denominator of 1.
Answers: Line
R: about (200-10)/2000 or .095/1, line S: about (80-30)/1500
or .033/1, line T: about (80-55)/1500 or .017/1)
- Does the steepest line always represent the greatest
total cost of copying? Provide examples to support your
answer.
Answer: No, for
instance R is the steepest line since it has the greatest
slope but it represents the least total cost when the
number of copies is less than 500. That is because, its
y-intercept is less than the y-intercepts of the other
lines.
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