Problem 5 - Understanding Elementary Algebra with Geometry

Question

The Lease-From-Us Company offers two different leasing plans for their top- ofthe-line color copying machine. The Economy plan costs \(\$ 175\) per month plus \(\$ 0.032\) per copy. The Standard plan costs \(\$ 225\) per month plus \(\$ 0.024\) per copy. (A) Write an equation for the monthly cost \(C\) of the Standard plan and the Economy for a month in which you make \(n\) copies. (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 copies per month make it more economical to buy the Economy plan.

Step-by-Step Solution

Answer

It's more economical to choose the Economy plan for fewer than \(6250\) copies per month.

Step 1: Formulate the Equation for Economy Plan

The Economy plan charges a fixed monthly fee of \(175\) dollars plus \(0.032\) dollars per copy. The equation for the monthly cost \(C_e\) is \[C_e = 175 + 0.032n\] where \(n\) is the number of copies.

Step 2: Formulate the Equation for Standard Plan

The Standard plan charges a fixed monthly fee of \(225\) dollars plus \(0.024\) dollars per copy. The equation for the monthly cost \(C_s\) is \[C_s = 225 + 0.024n\] where \(n\) is the number of copies.

Step 3: Sketch the Graph of Both Equations

Plot both equations \(C_e = 175 + 0.032n\) and \(C_s = 225 + 0.024n\) on a graph. The horizontal axis represents the number of copies \(n\) and the vertical axis represents the cost \(C\). The Economy plan starts at \(175\) with a steeper slope of \(0.032\), and the Standard plan starts at \(225\) with a slope of \(0.024\).

Step 4: Determine the Intersection Point

The point where the two graphs intersect is where the cost of the two plans is the same. Set the equations equal to each other to find \(n\): \[175 + 0.032n = 225 + 0.024n\] Simplify and solve for \(n\): \[175 + 0.032n - 0.024n = 225\] \[175 + 0.008n = 225\] \[0.008n = 50\] \[n = 6250\] Thus, the number of copies where both plans cost the same is \(6250\).

Step 5: Economical Plan Determination

For \(n < 6250\), the Economy plan is more economical because the total cost is less than that of the Standard plan. When \(n > 6250\), the Standard plan becomes more economical.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, representing relationships with a constant rate of change. Each linear equation can be described by the formula: \(y = mx + b\), where:

• \(y\) is the dependent variable (in this case, the monthly cost \(C\))
• \(x\) is the independent variable (the number of copies \(n\))
• \(m\) is the slope or rate of change
• \(b\) is the y-intercept, representing the starting value when \(x\) is zero

In this exercise, the linear equations for the Economy plan and Standard plan are: \(C_e = 175 + 0.032n\) and \(C_s = 225 + 0.024n\) These equations model the total monthly costs as functions of \(n\), the number of copies made. For the Economy plan, the slope is 0.032, indicating that each additional copy costs $ 0.032. The y-intercept is 175, the flat monthly fee. For the Standard plan, the slope is 0.024, and the y-intercept is 225.

Graphing

Graphing linear equations helps visualize the relationship between two variables. By plotting the equations of the Economy and Standard plans, you can see how costs change with the number of copies. To graph the equations:

• Set up a graph with the horizontal axis representing \(n\) (number of copies) and the vertical axis representing \(C\) (monthly cost).
• Plot the y-intercepts at 175 (Economy plan) and 225 (Standard plan).
• Use the slope values to determine the cost increase for each additional copy: 0.032 for the Economy plan and 0.024 for the Standard plan.
• Draw lines through these points to show the cost curves of each plan.

By observing these lines on the graph, you can identify where they intersect. The intersection point signifies the number of copies where both plans cost the same. In this case, the intersection occurs at 6,250 copies. Before this point, the Economy plan is cheaper. After this point, the Standard plan becomes more cost-effective.

Cost Comparison

Cost comparison analyzes which plan is more economical depending on the number of copies made. By solving the linear equations for their intersection, you can determine the breakpoint. To compare costs:

• Set the Economy equation equal to the Standard equation: \(175 + 0.032n = 225 + 0.024n\)
• Combine like terms: \(175 - 225 = 0.024n - 0.032n\)
• Simplify the equation to find \(n\): \(-50 = -0.008n\), so \(n = 6250\)

At 6250 copies, both plans cost the same. For fewer copies than 6250, the Economy plan is more economical. For more copies, the Standard plan saves money. This comparison helps businesses choose the most cost-effective plan based on their copying needs.