Problem 65 - Beginning and Intermediate Algebra

Question

During the school year, Tazia earns money by babysitting and tutoring. She can work at most \(15 \mathrm{hr}\) per week. Let \(x=\) number of hours Tazia babysits \(y=\) number of hours Tazia tutors a) Write a system of linear inequalities to describe the constraints on the number of hours Tazia can work per week. b) Graph the feasible region that describes how her hours can be distributed between babysitting and tutoring. c) Find three points in the feasible region and discuss their meanings. d) Find one point outside the feasible region and discuss its meaning.

Step-by-Step Solution

Answer

The constraints on Tazia's work hours can be represented by the system of linear inequalities: \(x + y \le 15\), \(x \ge 0\), and \(y \ge 0\), where \(x\) represents babysitting hours and \(y\) represents tutoring hours. The feasible region representing the possible distribution of her hours is the triangle formed by the intersection of the line \(x+y=15\) with the axes in the first quadrant. Examples of points within this feasible region include: (5,5), where Tazia evenly distributes her time between both activities; (0,10), where she only tutors; and (12,3), where she uses her maximum allowable hours, mostly for babysitting. A point outside the feasible region, such as (8,10), represents an impossible scenario where Tazia works more than her allowable weekly hours.

Step 1: (Step 1: Determine the constraints)

To set up the system of linear inequalities that describe the constraints on the number of hours Tazia can work per week: 1. She can work at most 15 hours per week: This implies the total time she spends on babysitting and tutoring should be equal to or less than 15 hours. That is: \(x + y \le 15\) 2. The number of hours she spends doing each activity should be non-negative: \(x\ge 0\) and \(y \ge 0\)

Step 2: (Step 2: Graph the feasible region)

To graph the feasible region, we first need to plot the given inequality constraints: 1. Plot the line \(x + y = 15\); The inequality \(x + y \le 15\) will be represented by the region below this line. 2. Plot the x-axis, which represents the inequality \(x \ge 0\). 3. Plot the y-axis, which represents the inequality \(y \ge 0\). The feasible region is the region where all these constraints are satisfied, which is the triangle formed by the intersection of the line \(x+y=15\) with the axes in the first quadrant.

Step 3: (Step 3: Discuss three points in the feasible region)

Choose three points within the feasible region and analyze their meaning: 1. Point A (5,5): This point indicates that Tazia spends 5 hours per week on babysitting and 5 hours per week on tutoring, for a total of 10 hours. She is working within her permissible hours and has evenly distributed her time between both activities. 2. Point B (0,10): This point indicates that Tazia spends no time on babysitting and 10 hours per week on tutoring. She is using her hours only on tutoring and is within her maximum allowed hours. 3. Point C (12,3): This point indicates that Tazia spends 12 hours per week on babysitting and 3 hours per week on tutoring, for a total of 15 hours. She is using her maximum allowable hours and devotes more time to babysitting.

Step 4: (Step 4: Discuss one point outside the feasible region)

Choose one point outside the feasible region and analyze its meaning: 1. Point D (8,10): This point indicates that Tazia spends 8 hours per week on babysitting and 10 hours per week on tutoring, for a total of 18 hours. Since she can work at most 15 hours per week, this point is not possible as it exceeds her allowable working hours.

Key Concepts

Feasible Region

The concept of a feasible region is crucial when solving problems involving linear inequalities. In Tazia's scenario, her weekly work schedule constraints define this feasible region. It's the section on a graph where all of her constraints are fulfilled. Think of it as the boundary inside which everything is possible when considering the constraints she must work with:- The first constraint is the maximum number of hours she can work per week, i.e., 15 hours.- The second constraint is that she cannot work negative hours.To visualize this:- Picture a graph with all possible hours plotted; those under the line formed by the equation \(x + y = 15\) and on or above the x and y axes form the feasible region.It's a triangular area that represents every possible combination of babysitting and tutoring hours that Tazia can work.

System of Equations

A system of equations gives us the rules or constraints within a problem that must be followed. In this context, Tazia’s system of linear inequalities:1. \(x + y \leq 15\) ensures she works no more than 15 hours per week.2. \(x \geq 0\) makes sure she doesn’t work negative babysitting hours.3. \(y \geq 0\) ensures non-negative tutoring hours.These equations, working together, set the limits for her weekly schedule. Solving these equations together graphically reveals which combinations of hours for babysitting and tutoring are possible. This approach helps in visualizing potential solutions and understanding the constraints clearly.

Graphing Inequalities

Graphing linear inequalities involves showing the solutions on a coordinate plane. It’s like painting regions that meet the criteria laid out by inequalities, like Tazia’s hours of work:1. Start by graphing the line \(x + y = 15\). Every point on this line adds up to exactly 15 hours. - The area below this line represents combinations under 15 hours.2. Draw the x- and y-axes as lines for \(x \ge 0\) and \(y \ge 0\) respectively, since she can't work less than zero hours.The overlapping area in the first quadrant (where both x and y are non-negative) gives us the feasible region.This visual method helps in quickly assessing possible schedules and choosing best options for time management.

Real-World Applications

Linear inequalities and feasible regions are not just theoretical concepts—they play a significant role in real-world applications like budgeting time or resources. For Tazia: - Her time constraints help simulate situations like planning work schedules or arranging timings efficiently. - Such methods are crucial in fields like finance, logistics, and project management. Why it matters: - These mathematical tools allow individuals to optimize their resources, like maximizing productivity or minimizing costs while adhering to set boundaries. - Understanding these concepts provides a structured approach to problem-solving in everyday situations. In essence, mastering such skills prepares one for using math practically to make informed decisions.