Thursday, July 16, 2020

Linear Equations Review - Word Problems - Practice Problems and Answers (#2)

Problems:

You have a choice of two cell phone companies, Ringer and Buzz.  Ringer charges 50 cents a phone call and 18 cents a minute.  Buzz charges 25 cents a phone call and 27 cents a minute.

a.  Write an equation for the cost of making a phone call using Ringer.  Write an equation for the cost of making a phone call using Buzz.

b.  Use the equations to find the point where the two lines intersect.  This is the exact problem as finding how many minutes you have to be on the phone for the two companies to charge the same, except you also have to find the cost.  Label this point on the graph.

c.  Graph both equations on the same set of axes.  Label your axes and choose an appropriate scale.  Only graph the portion that is relevant to the problem.

d.  Use the equations to determine when Ringer is more expensive than Buzz.

e.  Use the graph to determine when Buzz is more expensive than Ringer.

f.  What are the cost intercepts for Ringer and Buzz?

g.  What is the slope of each line?  What does the slope of each line mean in terms of the problem?

Answers:

a. Write an equation for the cost of making a phone call using Ringer. Write an equation for the cost of making a phone call using Buzz. Answer: Ringer: 0.18m + 0.50, Buzz: 0.27m + 0.25 b. Use the equations to find the point where the two lines intersect. This is the exact problem as finding how many minutes you have to be on the phone for the two companies to charge the same, except you also have to find the cost. Label this point on the graph. Answer: This amounts to solving 0.18m + 0.50 = 0.27m + 0.25. Thus, 0.18m + 0.50 = 0.27m + 0.25 implies that 0.50 = 0.09m + 0.25 implies that 0.25 = 0.09m. So, m = 0.25/0.09 = 25/9. Put m = 25/9 into either 0.18m + 0.50 or 0.27m + 0.25. Without loss of generality, we will put m = 25/9 into 0.18m + 0.50 to get 0.18(25/9) + 0.50 = 0.50 + 0.50 = 1. The point where the two lines intersect is (m, c) = (25/9, 1). We will label this point in the graph in c.

c. Graph both equation on the same set of axes . Label your axes and choose an appropriate scale. Only graph the portion that is relevant to the problem. Answer: The c-axis is the vertical axis and the m-axis is the horizontal axis. The line for Buzz starts at (0, 0.25). The line for Ringer starts at (0, 0.50). These lines intersect at the point (25/9, 1). d. Use the equations to determine when Ringer is more expensive than Buzz. Answer: Ringer is more expensive than Buzz when m is greater than or equal to 0 and less than 25/9. e. Use the graph to determine when Buzz is more expensive than Ringer. Buzz is more expensive than Ringer when m > 25/9. What are the cost-intercepts for Ringer and Buzz? Answer: This occurs when m = 0 in both equations. Thus, the cost intercept for Buzz is (m, c) = (0, 0.25) and the cost intercept for Ringer is (m, c) = (0, 0.50).

g. What is the slope of each line? What does the slope of each line mean in terms of this problem? Answer: The slope of the line for Ringer is 0.18. The slope of the line for Buzz is 0.27. The slope of each line is the cost per minute each company charges for a phone call.


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