Problem 47
Question
You are choosing between two cellphone plans. Data Plan A offers a flat monthly rate of \(\$ 20\) per gigabyte (GB). Data Plan B has a monthly fee of \(\$ 40\) with a charge of \(\$ 15\) per GB. For how many G.B of data will the costs for the two data plans be the same? What will be the cost for each plan? (Section 1.3 Example \(3)\)
Step-by-Step Solution
Verified Answer
The costs for the two data plans will be the same for 8 GBs of data, and the cost for each plan will be $160.
1Step 1: Setting Up the Equations
Start by translating the problem into mathematical equations. \nPlan A offers a flat rate of $20 per GB, which can be represented as \(y=20*GB\), where y is the total cost and GB is the gigabytes used. \nPlan B has a monthly fee of $40 plus $15 per GB, which can be represented as \(y=40+15*GB\). Our goal is to find the GB for which the costs of the two plans are the same, so we set the two equations equal to each other and solve for GB.
2Step 2: Solver the Equation
Let's solve the equation \(20*GB = 40 + 15*GB\). This simplifies to \(5*GB = 40\). Divide both sides by 5 to solve for GB: \(GB = 40/5\).
3Step 3: Verify the Solution
Plugging GB=8 back into both equations should give the same result. For Plan A, we have \(y=20*8 = 160\). For Plan B, we have \(y = 40 + 15*8 = 160\). The costs are indeed the same, which confirms our solution.
Key Concepts
Linear EquationsCost AnalysisProblem Solving
Linear Equations
Linear equations are essential tools in mathematics, especially when dealing with real-world problems like comparing cellphone plans. These equations model relationships between variables. In this context, we use linear equations to represent the cost of each cellphone plan based on data usage in gigabytes (GB).
For Plan A, the cost equation is represented as \( y = 20 \times \text{GB} \). This linear equation shows that the cost \( y \) increases linearly with the data usage (GB) at a rate of \(20 per GB.
Similarly, Plan B is represented by the equation \( y = 40 + 15 \times \text{GB} \), indicating a base cost of \)40 plus $15 for each GB. When these equations are set equal, they model the point where the costs of both plans intersect.
This intersection is crucial because it tells users at what data usage the plans are equally priced, helping in making an informed decision.
For Plan A, the cost equation is represented as \( y = 20 \times \text{GB} \). This linear equation shows that the cost \( y \) increases linearly with the data usage (GB) at a rate of \(20 per GB.
Similarly, Plan B is represented by the equation \( y = 40 + 15 \times \text{GB} \), indicating a base cost of \)40 plus $15 for each GB. When these equations are set equal, they model the point where the costs of both plans intersect.
This intersection is crucial because it tells users at what data usage the plans are equally priced, helping in making an informed decision.
Cost Analysis
Cost analysis involves understanding the financial implications of choosing between different options, crucial in our cellphone plan scenario. By translating the cost dynamics of each plan into equations, we can predict how costs will behave with different data usage levels.
Plan A, with a simple \( y = 20 \times \text{GB} \) model, is straightforward: each GB adds \(20 to the total monthly cost. This makes it easy to calculate the expense if you expect a certain data usage each month. You can quickly know if it fits your budget so you have no surprises.
Plan B, \( y = 40 + 15 \times \text{GB} \), on the other hand, begins with a higher fixed cost (\)40) but adds only $15 per GB. Understanding both equations allows consumers to see how their data needs translate to monthly costs and potentially save money. By comparing this figure with Plan A's cost at the same data usage, a user can determine which plan will be more economical.
Plan A, with a simple \( y = 20 \times \text{GB} \) model, is straightforward: each GB adds \(20 to the total monthly cost. This makes it easy to calculate the expense if you expect a certain data usage each month. You can quickly know if it fits your budget so you have no surprises.
Plan B, \( y = 40 + 15 \times \text{GB} \), on the other hand, begins with a higher fixed cost (\)40) but adds only $15 per GB. Understanding both equations allows consumers to see how their data needs translate to monthly costs and potentially save money. By comparing this figure with Plan A's cost at the same data usage, a user can determine which plan will be more economical.
Problem Solving
Problem solving is a critical skill that helps in finding solutions to real-life challenges, such as selecting the most cost-effective cellphone plan. The problem starts with identifying and translating the real-world situation into a mathematical model, using equations.
In this exercise, we set up the equations for both Plan A and Plan B and aimed to find the point of intersection. This method is analytical, allowing us to calculate precisely, rather than guesswork. Once the right number of gigabytes (GB) was found (GB = 8), we used substitution to verify our results in both equations, ensuring accuracy. This is an example of problem solving through the principle of verification.
The process teaches not only how to set up and solve equations but also strengthens analytical thinking. Such skills are essential for managing everyday decisions logically and effectively.
In this exercise, we set up the equations for both Plan A and Plan B and aimed to find the point of intersection. This method is analytical, allowing us to calculate precisely, rather than guesswork. Once the right number of gigabytes (GB) was found (GB = 8), we used substitution to verify our results in both equations, ensuring accuracy. This is an example of problem solving through the principle of verification.
The process teaches not only how to set up and solve equations but also strengthens analytical thinking. Such skills are essential for managing everyday decisions logically and effectively.
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Problem 47
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