/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Cellmate Communications offers t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 4

Cellmate Communications offers two monthly cellular phone plans. The Standard plan costs \(\$ 15\) per month plus \(\$ 0.22\) per minute of air time. The Deluxe plan costs \(\$ 35\) per month plus \(\$ 0.14\) per minute of air time. (A) Write an equation for the monthly cost \(C\) of the Standard plan and the Deluxe plan for a month in which you use \(m\) minutes. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(m\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many air time minutes per month make it more economical to buy the Standard plan.

Short Answer

Expert verified
The Standard plan is more economical for fewer than 250 minutes of air time per month.

Step by step solution

01

Formulate the Equation for the Standard Plan

The Standard plan costs a fixed monthly fee of \(15 plus \)0.22 per minute of air time. Therefore, the equation for the monthly cost, C, in terms of the number of minutes, m, is: \(C = 15 + 0.22m\)
02

Formulate the Equation for the Deluxe Plan

The Deluxe plan costs a fixed monthly fee of \(35 plus \)0.14 per minute of air time. Therefore, the equation for the monthly cost, C, in terms of the number of minutes, m, is: \(C = 35 + 0.14m\)
03

Sketch the Graphs of Both Plans

Plot the two equations on a graph with the horizontal axis representing the minutes, m, and the vertical axis representing the cost, C. The first line represents the Standard plan, and will start at \(15 on the vertical axis and rise with a slope of 0.22. The second line represents the Deluxe plan, starting at \)35 on the vertical axis with a slope of 0.14.
04

Determine the Intersection Point

To find the intersection point of the two lines, set the equations equal to each other: \(15 + 0.22m = 35 + 0.14m\)Solve for m by first isolating the terms with m: \(0.22m - 0.14m = 35 - 15\)\(0.08m = 20\)\(m = 250\)This means at 250 minutes, the cost for both plans will be equal.
05

Compare the Plans Based on the Intersection Point

For fewer than 250 minutes, the Standard plan will cost less since its variable cost per minute is higher but its fixed cost is lower. For more than 250 minutes, the Deluxe plan will cost less due to its lower per minute charge.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
Linear equations are equations of the first order. They represent relationships with constant rates of change.
In our problem, we have two cellular phone plans, each with its equation. The Standard plan has the equation for total monthly cost: \[C = 15 + 0.22m\] where \(C\) stands for the total cost and \(m\) represents the number of minutes used.
For the Deluxe plan, the equation is: \[C = 35 + 0.14m\]
Both equations are linear because they have the form: \[y = mx + b\] where \(m\) is the rate per minute (slope) and \(b\) is the fixed monthly fee (y-intercept). Understanding linear equations is crucial to analyzing and comparing these costs effectively.
graphing cost functions
Graphing cost functions helps visualize the relationships between minutes used and total cost.
We need to plot the equations for each plan on a graph with the horizontal axis representing minutes (\(m\)) and the vertical axis representing cost (\(C\)).
  • For the Standard plan, our line starts at 15 on the vertical axis and rises with a slope of 0.22.
  • For the Deluxe plan, our line starts at 35 on the vertical axis and rises with a slope of 0.14.
By graphing these, we can see at which point the costs are equal and where one plan becomes more economical over the other.
This visual representation is a powerful tool for comparing linear relationships.
solving systems of equations
When comparing two plans, finding their intersection point helps determine the break-even point.
To find when both plans cost the same, we set their equations equal to each other:
\[15 + 0.22m = 35 + 0.14m\]
Solving this system involves isolating \(m\):\[0.22m - 0.14m = 35 - 15\]
\[0.08m = 20\]
\[m = 250\]
At 250 minutes, both plans cost the same. To decide which plan to choose, compare their costs based on usage:
  • For less than 250 minutes, the Standard plan is cheaper.
  • For more than 250 minutes, the Deluxe plan is cheaper.
Understanding this allows for better decision-making based on individual usage.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Metropolitan Transportation Authority charges \(\$ 1.75\) per ride on public transportation. They offer a monthly commuter pass for \(\$ 48\) that allows unlimited travel on the public transportation system. Let \(n\) represent the number of trips taken per month on public transportation and \(C\) represent the cost of all these trips. (A) Write an equation for the transportation cost \(C\) if you buy the monthly pass and if you pay for each trip individually. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(n\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many trips per month make it more economical to buy a monthly pass rather than pay per trip.

Match the lettered items in column II with the numbered items in column I. Column I (1) Simultaneous equations in two variables with exactly one solution (2) An equation in one variable with no solutions (3) An equation in one variable with exactly one solution (4) Simultancous equations in two variables with infinitely many solutions (5) Simultaneous equations in two variables with no solutions (6) An equation in one variable with infinitely many solutions. Column II (a) Conditional (b) Consistent (c) Dependent (d) Inconsistent (e) Contradiction (f) Identity

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 5 x+2 y &=4 y+9 \\ y &=x-3 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Pat and Carlos both belong to the same book club. Pat orders two regular selections and three specially discounted ones for a total of \(\$ 56.90 .\) Carlos orders three regular selections and four specially discounted ones for a total of \(\$ 80.85\) What are the prices of a regular and a specially discounted selection?

Sketch a graph that represents the scenario described in the exercise. Be sure to clearly label any variables and the coordinate axes. Keep in mind that various graphs may be drawn to represent each situation. The price of a certain stock starts the day at \(\$ 15\) per share. Over the first 2 hours of trading, the price of the stock steadily declines to \(\$ 13\) per share. It remains at that price for 3 hours and then declines to \(\$ 11.50\) per share over the next hour.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.