Solving problems with tables and graphs
Objective: When done with this lesson, you will have demonstrated how to solve simple problems using tables and graphs.
Approximate completion time: 3 hours
Previously, in lesson 3.02, we looked at how to solve problems using tables. In this lesson we will expand on those ideas to solve more elaborate problems.
Consider the situation from assignment 5.01 regarding mowing services.
"Mike's Mowing Co. will mow lawns for a yearly fee of $50 and $12 per mowing. Carl's Cuts charges $120 for a yearly fee and then $6 per mowing. For example, to have your lawn mowed 2 times Mike's Mowing would cost 50 + 12·2 = $74 while Carl's Cuts would charge 120 + 6 ·2 = $132."
You were to do your own table, graph, and equations to model this scenario. Take a look at these tables.
# of mowings Mike's Mowing Charge ($) Carl's Cuts Charge ($) 0 50 120 1 62 126 2 74 132 3 86 138 4 98 144 5 110 150 Table 2
# of mowings Mike's Mowing Charge ($) Carl's Cuts Charge ($) 0 50 120 10 170 180 20 290 240 30 410 300 40 530 360 50 650 420
THINK DEEPER #1: Please open up a new word document and answer these questions about this lawn mowing scenario.
(Save the file as "assignment 5.02.yourname.doc" and keep it handy. Please copy and paste in the questions and remember to make your work stand out with something like blue font.)
- Explain the similarities in each table.
- Explain the differences in each table.
- Which table would Mike's Mowing Co. want to publish in their advertising? Explain.
- If someone had to choose between Mike's Mowing Co. and Carl's Cuts, what other information would you need to know before you could recommend which company would be better?
- If someone were to choose between the two companies to have their lawn mowed about 25 times this year, which company should they choose to minimize their costs? Please explain.
- Table 1 shows increments of 1 mowing. Table 2 shows increments of 10 mowings. Make your own table that would show increments of 5 mowings.
Now consider the following graphs that could be used to model the same situation.
THINK DEEPER #2: Please open up your "assignment 5.02.yourname.doc" to answer the following.
- Explain the similarities in each graph.
- Explain the differences in each graph.
- Which graph would Mike's Mowing Co. want to publish in their advertising? Explain.
- Use the graphs to estimate the total cost for mowing 18 lawns under each company. Explain how you arrived at your estimate.
- Approximate the coordinates of the point of intersection of the blue and green lines in the second graph. Explain the meaning of this point.
- What advise would you give someone interested in choosing between Mike's Mowing Co. and Carl's Cuts?
- If someone needed to know how much it would cost to have his or her lawn mowed 15 times, what model do you think would be easiest to use? Explain your reasoning.
With those tables and graphs to model the lawn mowing situation, nearly any question about pricing can be answered. Now look at another example.
Curly bought $325 worth of supplies to repair his fence. He's hired Handy Andy to work on the repairs. Andy charges $23 an hour.
- What would the total cost be for Curly if Andy works for 16 hours?
- How long could Andy work before Curly would the project would cost more than $500?
Solution with Table
Hours worked 0 2 4 6 8 10 12 14 16 Total Cost ($) 325 371 417 463 509 555 601 647 693
If Andy works for 16 hours, then the total cost for Curly would be $693.
If Andy works for 8 hours or more, then Curly's project would cost over $500.
Solution with Graph
Looking on the graph where the number of hours is 16, the cost is about $690. (The red dotted line shows that the actual cost is between $675 and $700, so any guess in that range is acceptable.) So if Andy works for 16 hours, then the total cost for Curly would be about $690.
Looking at the graph where the total cost is $500, the number of hours is a little less than 8. (The blue dotted line shows that the actual number of hours is between 6 and 8. Although any number between 6 and 8 is acceptable, it would be better to estimate closer to 8.) So if Andy works for 8 hours or more, then Curly's project would cost over $500.
Note that solutions by table or graph require a conclusion statement.
THINK DEEPER #3: Please open up your "assignment 5.02.yourname.doc" to answer the following.
- Explain the similarities between the tabular and graphical solutions.
- Explain the differences between the tabular and graphical solutions.
- Explain why the table solution would not work as well if the question had been "... if Andy works for 15 hours".
- Explain how to use the table to answer "What would the total cost be for Curly if Andy works for 20 hours?"
- Explain how to use the graph to answer "What would the total cost be for Curly if Andy works for 20 hours?"
- The "starting point" for Curly's costs was $325. A good table would begin with such a starting point. Where is this starting point on the graph?
In your "assignment 5.02.yourname.doc"
- Include your THINK DEEPER repsonses.
- Make a table and a graph to solve each of the following problems:
- Cal has $20 in savings. Cindy has $0 in savings. Cal starts saving $10 a week and Cindy starts saving $15 a week. How many weeks until Cindy has more money than Cal?
- Ted and Rhonda are racing. Ted has 30 marshmallows left to eat while Rhonda has only 20 left. Ted is eating them at a rate of 3 every minute. Rhonda can eat only 2 a minute. Who will finish first?
- Mike is thinking of buying a car. The 1973 Datsun will cost him $300. However, it will cost him about $150 a month in repairs. The 1985 Volkswagen will cost him $900, but it probably will only need about $25 a month in repairs. How many months until the Datsun will cost him just as much as the Volkswagen?