Problem 6 - Understanding Elementary Algebra with Geometry

Question

An appliance store offers two lease plans for renting a refrigerator. Plan A costs a Elat fee of \(\$ 75\) plus a monthly rental fee of \(\$ 28 .\) Plan \(B\) costs a flat fee of \(\$ 50\) Slus a monthly rental fee of \(\$ 34\). (A) Write an equation for the total cost \(C\) of renting a refrigerator under both plans for \(m\) months. (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 when each plan is more economical.

Step-by-Step Solution

Answer

The total cost equations are: Plan A: \( C_A = 75 + 28m \); Plan B: \( C_B = 50 + 34m \). Plan A is more economical when renting for more than about 4.17 months.

Step 1 - Write the equations for Plan A and Plan B

For Plan A, the total cost is given by a flat fee plus the monthly rental fee multiplied by the number of months. Hence, the equation for Plan A is: Plan A: \[ C_A = 75 + 28m \]For Plan B, the total cost is similar – a flat fee plus the monthly rental fee multiplied by the number of months. Hence, the equation for Plan B is: Plan B: \[ C_B = 50 + 34m \]

Step 2 - Sketch the graphs of the two equations

To graph the equations obtained in Step 1, plot the cost for different values of months (m) for both plans on the same set of axes.1. For Plan A: When m = 0, \(C_A = 75\). When m = 1, \(C_A = 75 + 28(1) = 103\), and so on.2. For Plan B: When m = 0, \(C_B = 50\). When m = 1, \(C_B = 50 + 34(1) = 84\), and so on.

Step 3 - Determine when each plan is more economical

To find where each plan is more economical, set the costs of both plans equal: \[ 75 + 28m = 50 + 34m \]Solve for m: \[ 75 - 50 = 34m - 28m \]\[ 25 = 6m \]\[ m = \frac{25}{6} \approx 4.17 \]For m < 4.17, Plan B is more economical. For m > 4.17, Plan A is more economical.

Key Concepts

linear equations

Linear equations are mathematical expressions that represent a straight line when graphed. They are usually written in the form of: \( y = mx + b \), where:

• \( y \) is the dependent variable (total cost in our case)
• \( m \) is the slope (rate of change, or monthly cost)
• \( x \) is the independent variable (number of months)
• \( b \) is the y-intercept (the starting cost or flat fee)

In the context of the exercise:

• For Plan A: \( C_A = 75 + 28m \)
• For Plan B: \( C_B = 50 + 34m \)

Here, \( C_A \) and \( C_B \) are the total costs, \(75\) and \(50\) are flat fees, and \(28\) and \(34\) are the monthly rental fees. We'll use these equations to compare leasing plans. Remember, the goal is to determine which plan is cheaper over different durations by solving and graphing these linear equations.

graphing

Graphing linear equations helps visually compare different plans. In this scenario, you will graph \( C_A = 75 + 28m \) and \( C_B = 50 + 34m \) on the same set of axes.

To graph:

• Start with the y-intercept, where \(m=0\). For Plan A, it's \(75\), and for Plan B, it's \(50\).
• Then, plot more points by calculating the total cost for different months, eg., \(m=1\), \(m=2\), etc.
• Draw lines connecting these points for each plan.

The points where these lines intersect and change relationships reveal the more economical plan depending on the lease duration. From the graph, identify the intersection which shows the switch-over point.

cost analysis

Cost analysis helps determine which plan saves more money over time. By comparing the equations:

• Plan A: \( C_A = 75 + 28m \)
• Plan B: \( C_B = 50 + 34m \)

Both flat fees and monthly costs need attention. Set the equations equal to find the breakeven point: \(75 + 28m = 50 + 34m\)

Simplify to solve for \(m\):

• \(75 - 50 = 34m - 28m\)
• \(25 = 6m\)
• \(m = \frac{25}{6} \approx 4.17\)

This means, Plan B is cheaper for less than 4.17 months, and Plan A is cheaper beyond 4.17 months.

comparison of plans

Comparing plans involves examining the total cost for various durations. For example, for both Plan A and Plan B:

• Find the cost at different months: 1 month, 2 months, 3 months, etc.
• Identify the max length you may rent when estimating costs.

Analyze where the lines cross on your graph. This cross-point, or breakeven point, is where both plans cost the same.

From earlier, we found this is around 4.17 months. So:

• For fewer than 4.17 months, Plan B is cheaper.
• For more than 4.17 months, Plan A will cost less.

Use this information for real-life decisions on lease plans. Always think about your specific needs and durations to choose the most cost-effective option. It simplifies planning and saves money!