Problem 40 - Algebra 1: Concepts and Skills

Question

A local computer center charges nonmembers \(\$5\) per session to use the media center. Members are charged a one-time fee of \(\$20\) and \(\$3\) per session. Use the verbal model to write an equation that can help you decide whether to become a member. Solve the equation and explain your solution.

Step-by-Step Solution

Answer

It is more economical to become a member if you are planning to use the media center for more than 10 sessions. \( x = 10 \) is the break-even point where the costs for membership and non-membership are equal.

Step 1: Define the Cost Functions

Start by defining the two cost functions. For a nonmember, the cost function is \( C_{nm}(x) = 5x \), where \( x \) is the number of sessions. For a member, the cost function is \( C_m(x) = 20 + 3x \).

Step 2: Set Up the Equality

Next, set up an equality to find out for which number of sessions both options would cost the same. This gives the equation: \( C_{nm}(x) = C_m(x) \). Substitute the functions from Step 1 into this equation to get \( 5x = 20 + 3x \).

Step 3: Solve for x

Solve the equation for \( x \), which represents the number of sessions. Subtract \( 3x \) from both sides to get \( 2x = 20 \). Then, divide by 2 on both sides to solve for \( x \) to get \( x = 10 \).

Step 4: Interpret the Result

The solution \( x = 10 \) represents the number of sessions at which both options (membership and non-membership) will cost the same. Therefore, if you are planning to use the media center for more than 10 sessions, it would be more economical to become a member.

Key Concepts

Cost Analysis

Cost analysis is a process used to evaluate the expenses associated with different choices. In this exercise, we are comparing two options: using the media center as a nonmember or becoming a member. To conduct a cost analysis, we first need to establish the costs associated with each choice.

• Nonmember Cost: The cost function for using the media center as a nonmember is defined by the equation \( C_{nm}(x) = 5x \). Here, \( x \) stands for the number of sessions. This means that for every session, a nonmember pays \( \\(5 \).
• Member Cost: For members, the cost function is \( C_m(x) = 20 + 3x \). Members pay a one-time fee of \( \\)20 \) plus \( \$3 \) per session. The initial cost includes a fixed fee, making the member rate change more dependent on the number of sessions attended.

By analyzing these cost functions, you can determine how many sessions must be attended before the membership option becomes less expensive than the nonmember option.

Membership Decision

Choosing whether to become a member involves analyzing the costs and benefits of each option. In this context, a rational decision considers the point at which the cost of being a nonmember equals the cost of being a member. A key aspect of making this decision is setting up an equation based on the cost functions established. The equality \( C_{nm}(x) = C_m(x) \) is used to discern how many sessions are needed before membership becomes more advantageous.

• Set Up the Equation: Substitute the known cost functions into the equality: \( 5x = 20 + 3x \).
• Solve the Equation: Simplifying this equation allows us to solve for \( x \), the number of sessions. By rewriting it to \( 5x - 3x = 20 \), we find \( 2x = 20 \), leading to \( x = 10 \).

This calculation guides the membership decision. If you plan on attending more than 10 sessions, a membership is the more economical option.

Linear Functions

Linear functions are mathematical expressions used to describe linear relationships. Each function generates a straight line when plotted on a graph, showing how variables correspond with one another. In this exercise, the cost functions for members and nonmembers are linear.

• Definition: A linear function can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• Nonmember Linear Function: In \( C_{nm}(x) = 5x \), there is no intercept term \( b \), which depicts that the line passes through the origin. The slope, \( m = 5 \), indicates the cost increase per session.
• Member Linear Function: \( C_m(x) = 20 + 3x \) shows a slope of \( m = 3 \) and an intercept \( b = 20 \), representing the initial flat fee.

These functions give insight into how long-term costs evolve with increasing session numbers. By graphing these equations, their intersection at \( x = 10 \) visually confirms where the expense is balanced for both options.