4.11: Problem Solving with Linear Graphs
What if you borrowed $2000 from your parents for a used car? At the end of the summer you pay them back $750 that you earned from your summer job. Throughout the school year, you will pay them another $125 per month from your part-time job. How many months will it take you to pay them back? After completing this Concept, you'll be able to solve problems like this one using graphs.
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CK-12 Foundation: 0411S Problem-Solving Strategies - Graphs (H264)
Guidance
In this chapter, we’ve been solving problems where quantities are linearly related to each other. In this section, we’ll look at a few examples of linear relationships that occur in real-world problems, and see how we can solve them using graphs. Remember back to our Problem Solving Plan:
- Understand the Problem
- Devise a Plan—Translate
- Carry Out the Plan—Solve
- Look—Check and Interpret
Example A
A cell phone company is offering its costumers the following deal: You can buy a new cell phone for $60 and pay a monthly flat rate of $40 per month for unlimited calls. How much money will this deal cost you after 9 months?
Solution
Let’s follow the problem solving plan.
Step 1: The phone costs $60; the calling plan costs $40 per month.
Let \begin{align*}x = \end{align*} number of months.
Let \begin{align*}y = \end{align*} total cost.
Step 2: We can solve this problem by making a graph that shows the number of months on the horizontal axis and the cost on the vertical axis.
Since you pay $60 when you get the phone, the \begin{align*}y-\end{align*}intercept is (0, 60).
You pay $40 for each month, so the cost rises by $40 for 1 month, so the slope is 40.
We can graph this line using the slope-intercept method.
Step 3: The question was “How much will this deal cost after 9 months?” We can now read the answer from the graph. We draw a vertical line from 9 months until it meets the graph, and then draw a horizontal line until it meets the vertical axis.
We see that after 9 months you pay approximately $420.
Step 4: To check if this is correct, let’s think of the deal again.
Originally, you pay $60 and then $40 a month for 9 months.
\begin{align*}\text{Phone} & = \$60\\ \text{Calling plan} & = \$40 \times 9 = \$360\\ \text{Total cost} & = \$420.\end{align*}
The answer checks out.
Example B
A stretched spring has a length of 12 inches when a weight of 2 lbs is attached to the spring. The same spring has a length of 18 inches when a weight of 5 lbs is attached to the spring. What is the length of the spring when no weights are attached?
Solution
Step 1: We know: the length of the spring = 12 inches when weight = 2 lbs
the length of the spring = 18 inches when weight = 5 lbs
We want: the length of the spring when weight = 0 lbs
Let \begin{align*}x = \end{align*} the weight attached to the spring.
Let \begin{align*}y = \end{align*} the length of the spring.
Step 2: We can solve this problem by making a graph that shows the weight on the horizontal axis and the length of the spring on the vertical axis.
We have two points we can graph:
When the weight is 2 lbs, the length of the spring is 12 inches. This gives point (2, 12).
When the weight is 5 lbs, the length of the spring is 18 inches. This gives point (5, 18).
Graphing those two points and connecting them gives us our line.
Step 3: The question was: “What is the length of the spring when no weights are attached?
We can answer this question by reading the graph we just made. When there is no weight on the spring, the \begin{align*}x-\end{align*}value equals zero, so we are just looking for the \begin{align*}y-\end{align*}intercept of the graph. On the graph, the \begin{align*}y-\end{align*}intercept appears to be approximately 8 inches.
Step 4: To check if this correct, let’s think of the problem again.
You can see that the length of the spring goes up by 6 inches when the weight is increased by 3 lbs, so the slope of the line is \begin{align*}\frac{6 \ inches}{3 \ lbs} = 2 \ inches/lb\end{align*}.
To find the length of the spring when there is no weight attached, we can look at the spring when there are 2 lbs attached. For each pound we take off, the spring will shorten by 2 inches. If we take off 2 lbs, the spring will be shorter by 4 inches. So, the length of the spring with no weights is 12 inches - 4 inches = 8 inches.
The answer checks out.
Example C
Christine took 1 hour to read 22 pages of Harry Potter. She has 100 pages left to read in order to finish the book. How much time should she expect to spend reading in order to finish the book?
Solution
Step 1: We know - Christine takes 1 hour to read 22 pages.
We want - How much time it takes to read 100 pages.
Let \begin{align*}x = \end{align*} the time expressed in hours.
Let \begin{align*}y = \end{align*} the number of pages.
Step 2: We can solve this problem by making a graph that shows the number of hours spent reading on the horizontal axis and the number of pages on the vertical axis.
We have two points we can graph:
Christine takes 1 hour to read 22 pages. This gives point (1, 22).
A second point is not given, but we know that Christine would take 0 hours to read 0 pages. This gives point (0, 0).
Graphing those two points and connecting them gives us our line.
Step 3: The question was: “How much time should Christine expect to spend reading 100 pages?” We can find the answer from reading the graph - we draw a horizontal line from 100 pages until it meets the graph and then we draw the vertical until it meets the horizontal axis. We see that it takes approximately 4.5 hours to read the remaining 100 pages.
Step 4: To check if this correct, let’s think of the problem again.
We know that Christine reads 22 pages per hour - this is the slope of the line or the rate at which she is reading. To find how many hours it takes her to read 100 pages, we divide the number of pages by the rate. In this case, \begin{align*}\frac{100 \ pages}{22 \ pages/hour} = 4.54 \ hours\end{align*}. This is very close to the answer we got from reading the graph.
The answer checks out.
Watch this video for help with the Examples above.
CK-12 Foundation: Problem Solving Strategies- Graphs
Vocabulary
The four steps of the problem solving plan when using graphs are:
- Understand the Problem
- Devise a Plan—Translate: Make a graph.
- Carry Out the Plan—Solve: Use the graph to answer the question asked.
- Look—Check and Interpret
Guided Practice
Aatif wants to buy a surfboard that costs $249. He was given a birthday present of $50 and he has a summer job that pays him $6.50 per hour. To be able to buy the surfboard, how many hours does he need to work?
Solution
Step 1: We know - The surfboard costs $249.
Aatif has $50.
His job pays $6.50 per hour.
We want - How many hours Aatif needs to work to buy the surfboard.
Let \begin{align*}x = \end{align*} the time expressed in hours
Let \begin{align*}y = \end{align*} Aatif’s earnings
Step 2: We can solve this problem by making a graph that shows the number of hours spent working on the horizontal axis and Aatif’s earnings on the vertical axis.
Aatif has $50 at the beginning. This is the \begin{align*}y-\end{align*}intercept: (0, 50).
He earns $6.50 per hour. This is the slope of the line.
We can graph this line using the slope-intercept method. We graph the \begin{align*}y-\end{align*}intercept of (0, 50), and we know that for each unit in the horizontal direction, the line rises by 6.5 units in the vertical direction. Here is the line that describes this situation.
Step 3: The question was: “How many hours does Aatif need to work to buy the surfboard?”
We find the answer from reading the graph - since the surfboard costs $249, we draw a horizontal line from $249 on the vertical axis until it meets the graph and then we draw a vertical line downwards until it meets the horizontal axis. We see that it takes approximately 31 hours to earn the money.
Step 4: To check if this correct, let’s think of the problem again.
We know that Aatif has $50 and needs $249 to buy the surfboard. So, he needs to earn \begin{align*}\$249 - \$50 = \$199\end{align*} from his job.
His job pays $6.50 per hour. To find how many hours he need to work we divide: \begin{align*}\frac{\$199}{\$6.50/hour} = 30.6 \ hours\end{align*}. This is very close to the answer we got from reading the graph.
The answer checks out.
Practice
Solve the following problems by making a graph and reading it.
- A gym is offering a deal to new members. Customers can sign up by paying a registration fee of $200 and a monthly fee of $39.
- How much will this membership cost a member by the end of the year?
- The old membership rate was $49 a month with a registration fee of $100. How much more would a year’s membership cost at that rate?
- Bonus: For what number of months would the two membership rates be the same?
- A candle is burning at a linear rate. The candle measures five inches two minutes after it was lit. It measures three inches eight minutes after it was lit.
- What was the original length of the candle?
- How long will it take to burn down to a half-inch stub?
- Six half-inch stubs of candle can be melted together to make a new candle measuring \begin{align*}2\frac{5}{6}\end{align*} inches (a little wax gets lost in the process). How many stubs would it take to make three candles the size of the original candle?
- A dipped candle is made by taking a wick and dipping it repeatedly in melted wax. The candle gets a little bit thicker with each added layer of wax. After it has been dipped three times, the candle is 6.5 mm thick. After it has been dipped six times, it is 11 mm thick.
- How thick is the wick before the wax is added?
- How many times does the wick need to be dipped to create a candle 2 cm thick?
- Tali is trying to find the thickness of a page of his telephone book. In order to do this, he takes a measurement and finds out that 55 pages measures \begin{align*}\frac{1}{8} \ inch\end{align*}. What is the thickness of one page of the phone book?
- Bobby and Petra are running a lemonade stand and they charge 45 cents for each glass of lemonade. In order to break even they must make $25.
- How many glasses of lemonade must they sell to break even?
- When they’ve sold $18 worth of lemonade, they realize that they only have enough lemons left to make 10 more glasses. To break even now, they’ll need to sell those last 10 glasses at a higher price. What does the new price need to be?
- Dale is making cookies using a recipe that calls for 2.5 cups of flour for two dozen cookies. How many cups of flour does he need to make five dozen cookies?
- To buy a car, Jason makes a down payment of $1500 and pays $350 per month in installments.
- How much money has Jason paid at the end of one year?
- If the total cost of the car is $8500, how long will it take Jason to finish paying it off?
- The resale value of the car decreases by $100 each month from the original purchase price. If Jason sells the car as soon as he finishes paying it off, how much will he get for it?
- Anne transplants a rose seedling in her garden. She wants to track the growth of the rose so she measures its height every week. On the third week, she finds that the rose is 10 inches tall and on the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, what was the height of the rose when Anne planted it?
- Ravi hangs from a giant spring whose length is 5 m. When his child Nimi hangs from the spring its length is 2 m. Ravi weighs 160 lbs and Nimi weighs 40 lbs. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs, hangs from it?
- Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is 20 cm and for a 300 gram weight the length of the stretched spring is 25 cm.
- What is the unstretched length of the spring?
- If the spring can only stretch to twice its unstretched length before it breaks, how much weight can it hold?
- Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet.
- What was the submarine’s depth five minutes after it started surfacing?
- How much longer would it take at that rate to get all the way to the surface?
- Kiersta’s phone has completely run out of battery power when she puts it on the charger. Ten minutes later, when the phone is 40% recharged, Kiersta’s friend Danielle calls and Kiersta takes the phone off the charger to talk to her. When she hangs up 45 minutes later, her phone has 10% of its charge left. Then she gets another call from her friend Kwan.
- How long can she spend talking to Kwan before the battery runs out again?
- If she puts the phone back on the charger afterward, how long will it take to recharge completely?
- Marji is painting a 75-foot fence. She starts applying the first coat of paint at 2 PM, and by 2:10 she has painted 30 feet of the fence. At 2:15, her husband, who paints about \begin{align*}\frac{2}{3}\end{align*} as fast as she does, comes to join her.
- How much of the fence has Marji painted when her husband joins in?
- When will they have painted the whole fence?
- How long will it take them to apply the second coat of paint if they work together the whole time?
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9614.
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Here you'll learn how to follow a problem-solving plan that includes graphing to solve real-world problems.