Two cell phone companies have different rate plans. Runfast has monthly charges $25 plus $10 per gig of data. B A &D’s monthly charge is $18 plus 15 per gig of data. Your task is to determine under what circumstances each company has the better pricing. You will derive and solve a set of equations, graph those equations, and evaluate the meanings for these events. Don’t worry; I have a series of questions to guide you through the process. Good luck!
Determine the equation for the monthly charges for Runfast and B A&D.
Runfast= _25+(10*x)____________
BA&D=___18+(15*x)_________
Use your mathematical skill to solve this system of equations using the substitution method. (show your work here)
Graph the results on the accompanying graph as a visual aid. Don’t forget to label which line is for which company.
Why don’t the lines start at the origin?_______________________________________________
What is special about each of the y-intercepts?________________________________________
Which company has the better deal for customers that don’t us much data? How do you know?___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
At what point are the plans the same amount? How do you know?___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Which company has the better deal for customers that use a lot of data? How do you know?___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
They gave you the functions. Did you graph them as suggested?
Here are the lines:
http://www.wolframalpha.com/input/?i=plot+y%3D25%2B10x+,+y+%3D+18%2B15x
Now you can surely answer the questions.
To solve this problem, let's first determine the equations for the monthly charges for Runfast and BA&D.
The equation for the monthly charges for Runfast is: Runfast = $25 + $10 * x, where "x" represents the number of gigabytes of data used.
The equation for the monthly charges for BA&D is: BA&D = $18 + $15 * x.
To solve this system of equations using the substitution method, we can set the two equations equal to each other and solve for "x".
$25 + $10 * x = $18 + $15 * x
Simplifying the equation:
$7 = $5 * x
Dividing both sides of the equation by $5:
x = $7/$5
x = 1.4
Therefore, when x = 1.4 (or 1.4 gigabytes of data), the monthly charges for both companies are equal.
Now, let's graph the results on an accompanying graph.
The graph should have the monthly charges on the y-axis and the number of gigabytes of data used on the x-axis. Label one line as "Runfast" and the other line as "BA&D."
Now let's address the additional questions:
1. Why don’t the lines start at the origin?
The lines don't start at the origin because both companies have a base monthly charge before accounting for the additional charges per gigabyte of data. The starting point of each line represents the base monthly charge.
2. What is special about each of the y-intercepts?
The y-intercept represents the monthly charge when no data is used (x = 0). For Runfast, the y-intercept is $25, which means there is a $25 monthly charge irrespective of data usage. For BA&D, the y-intercept is $18, which means there is an $18 monthly charge irrespective of data usage.
3. Which company has the better deal for customers that don't use much data? How do you know?
Runfast has the better deal for customers who don't use much data because their monthly charge is lower at lower data usage levels. As the number of gigabytes used approaches zero, the Runfast line approaches its y-intercept of $25, which is lower than BA&D's y-intercept of $18.
4. At what point are the plans the same amount? How do you know?
The plans are the same amount when x = 1.4 (or 1.4 gigabytes of data). This means that if the number of gigabytes used is 1.4, both Runfast and BA&D will have the same monthly charges. We found this point of intersection by solving the system of equations.
5. Which company has the better deal for customers that use a lot of data? How do you know?
BA&D has the better deal for customers who use a lot of data because their additional charge per gigabyte is lower than Runfast's. As the number of gigabytes used increases, the slope of the BA&D line ($15) is lower than the slope of the Runfast line ($10), indicating that BA&D's monthly charges increase at a slower rate for higher data usage.