4.8: Problem-Solving Strategies - Graphs
Learning Objectives
- Read and understand given problem situations.
- Use the strategy: read a graph.
- Develop and apply the strategy: make a graph.
- Solve real-world problems using selected strategies as part of a plan.
Introduction
In this chapter, we have been solving problems where quantities are linearly related to each other. In this section, we will look at a few examples of linear relationships that occur in real-world problems. Remember back to our Problem Solving Plan.
Step 1:
Understand the problem
Read the problem carefully. Once the problem is read, list all the components and data that are involved. This is where you will be assigning your variables.
Step 2:
Devise a plan - Translate
Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart or construct a table as a start to solving your problem.
Step 3:
Carry out the plan - Solve
This is where you solve the equation you came up with in Step 2.
Step 4:
Look - Check and Interpret
Check to see if your answer makes sense.
Let’s look at an example that investigates a geometrical relationship.
Example 1
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:
\begin{align*}\text{cell phone} = \$60, \text{calling plan} = \$40 \ \text{per month}\! \\ \text{Let} \ x = \text{number of months}\! \\ \text{Let} \ y = \text{cost in dollars}\end{align*}
Step 2: Let’s 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 for the phone when you get the phone, then the \begin{align*}y-\end{align*}intercept is (0, 60).
You pay $40 for each month, so the cost rises by $40 for one month, so the slope = 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 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 2
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. It is known from physics that within certain weight limits, the function that describes how much a spring stretches with different weights is a linear function. What is the length of the spring when no weights are attached?
Solution
Let’s apply problem solving techniques to our problem.
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 the 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
Let’s 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).
If we join these two points by a line and extend it in both directions we get the relationship between weight and length of the spring.
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 to zero, so we are just looking for the \begin{align*}y-\end{align*}intercept of the graph. Looking at the graph we see that the \begin{align*}y-\end{align*}intercept is 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 look at the spring when there are 2 lbs attached. For each pound we take off, the spring will shorten by 2 inches. Since we take off 2 lbs, the spring will be shorter by 4 inches. So, the length of the spring with no weights is \begin{align*}12\ inches \ -4\ inches = 8\ inches\end{align*}.
The answer checks out.
Example 3
Christine took one hour to read 22 pages of Harry Potter and the Order of the Phoenix. She has 100 pages left to read in order to finish the book. Assuming that she reads at a constant rate of pages per hour, how much time should she expect to spend reading in order to finish the book?
Solution: Let’s apply the problem solving techniques:
Step 1
We know that it takes Christine takes 1 hour to read 22 pages.
We want to know how much time it takes her 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
Let’s 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 one hour to read 22 pages. This gives point (1, 22).
A second point is not given but we know that Christine takes 0 hours to read 0 pages. This gives the point (0, 0).
If we join these two points by a line and extend it in both directions we get the relationship between the amount of time spent reading and the number of pages read.
Step 3
The question was: “How much time should Christine expect to spend reading 100 pages?”
We 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 \ \text{pages}}{22 \ \text{pages per hour}} =4.54 \ \text{hours}\end{align*}. This is very close to what we gathered from reading the graph.
The answer checks out.
Example 4
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
Let’s apply the problem solving techniques.
Step 1
We know - Surfboard costs $249.
He has $50.
His job pays $6.50 per hour.
We want - How many hours does Aatif need 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
Let’s 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.
Peter has $50 at the beginning. This is the \begin{align*}y-\end{align*}intercept of (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 in order 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\ \text{per hour}} = 30.6\ \text{hours}\end{align*}. This is very close to the result we obtained from reading the graph.
The answer checks out.
Lesson Summary
The four steps of the problem solving plan are:
- Understand the problem
- Devise a plan - Translate. Build a graph.
- Carry out the plan - Solve. Use the graph to answer the question asked.
- Look - Check and Interpret
Review Questions
Solve the following problems by making a graph and reading a graph.
- 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?
- 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?
- Tali is trying to find the width of a page of his telephone book. In order to do this, he takes a measurement and finds out that 550 pages measures 1.25 inches. What is the width 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?
Review Answers
- $668
- 5.67 inches
- 0.0023 inches
- 56 glasses
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.