91Ó°ÊÓ

You are the marketing director for a company that manufactures bodybuilding supplements and you are planning to run ads in Sports Illustrated and \(G Q\) Magazine. Based on readership data, you estimate that each one-page ad in Sports Illustrated will be read by 650,000 people in your target group, while each one-page ad in \(G Q\) will be read by \(150,000 .{ }^{9}\) You would like your ads to be read by at least three million people in the target group and, to ensure the broadest possible audience, you would like to place at least three fullpage ads in each magazine. Draw the feasible region that shows how many pages you can purchase in each magazine. Find the corner points of the region. (Round each coordinate to the nearest whole number.)

Step by step solution

01

Define Variables

Let's define the variables: - Let x be the number of one-page ads in Sports Illustrated - Let y be the number of one-page ads in GQ Magazine

02

Write Down the Constraints

According to the problem, we have the following constraints: 1. At least three million target readers: \(650,000x + 150,000y \geq 3,000,000\) 2. At least three full-page ads in each magazine: \(x \geq 3\), \(y \geq 3\)

03

Sketch the Feasible Region

In order to draw the feasible region, we need to sketch the inequalities on a coordinate plane (x, y). For each constraint: 1. \(650,000x + 150,000y \geq 3,000,000\) => \(y \geq \frac{3,000,000 - 650,000x}{150,000}\) 2. \(x \geq 3\) 3. \(y \geq 3\) For each inequality, sketch the region where the inequality is satisfied (above/below/right/left the line). Then, overlay the graphs of all three inequalities and find the intersection of all three regions. This intersection is the feasible region.

04

Find the corner points of the feasible region

To find the corner points of the feasible region, we need to identify the intersection points of the lines from step 3 or any points where the inequality changes. The corner points can be found by solving the systems of linear equations made up from the boundaries of the inequalities: 1. Intersection between constraints 1 and 2: Solve \(650,000x + 150,000y = 3,000,000\) \(x = 3\) 2. Intersection between constraints 1 and 3: Solve \(650,000x + 150,000y = 3,000,000\) \(y = 3\) 3. Intersection between constraints 2 and 3: The point (3, 3) Once the coordinates of the corner points are found, round each coordinate to the nearest whole number.

05

List the corner points

Write down the corner points you found in step 4, rounded to the nearest whole number. The exact corner points are (3, 15), (19, 3), and (3, 3). The rounded corner points are as follows: 1. (3, 15) 2. (19, 3) 3. (3, 3) These are the corner points of the feasible region that represent the possible number of one-page ads in Sports Illustrated and \(GQ\) Magazine considering the given constraints.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Feasible Region

When solving a linear programming problem, the feasible region is a crucial concept. It represents all possible solutions that satisfy the given constraints. In our marketing exercise, this region reflects the combinations of ads in Sports Illustrated and GQ Magazine, which meet the business goals set by the marketing director.

To find the feasible region, we first graph the inequalities derived from the constraints. Each inequality defines a half-plane, and the feasible region is the overlapping area where all conditions are simultaneously met. This region on the graph shows every viable option for how many ads can be placed in each magazine.

• This region can be bounded or unbounded, being a polygon or an open region, determined by the nature of the constraints.
• In our problem, the feasible region is a polygon defined by straight lines from the inequalities.

Constraints

Constraints are the conditions or limits that must be satisfied in a linear programming problem. They are usually expressed as linear inequalities. For our exercise, the constraints ensure that our advertising reaches enough people while maintaining a minimum number of ads in each magazine.

In this particular problem, we have three constraints:

• The total readership must reach at least three million people, expressed by the inequality: \(650,000x + 150,000y \geq 3,000,000\).
• There must be at least three ads in Sports Illustrated: \(x \geq 3\).
• Similarly, at least three ads must be placed in GQ Magazine: \(y \geq 3\).

These constraints construct boundaries within which the feasible region is defined. They guide the graphing of the problem, helping to find solutions that satisfy all necessary conditions.

Corner Points

Corner points of the feasible region are the vertices of the polygon created by the intersecting lines of the constraints. These points are significant because, in linear programming, the optimal solution often lies at one of these vertices.

To find these points, we look at where the boundary lines intersect. In our exercise:

• Solving the system of equations formed by \(650,000x + 150,000y = 3,000,000\) and \(x = 3\), we find one intersection.
• Another intersection is from \(650,000x + 150,000y = 3,000,000\) and \(y = 3\).
• The point \((3, 3)\) is found where the constraints \(x \geq 3\) and \(y \geq 3\) meet.

Rounded to the nearest whole number, the corner points are \((3, 15)\), \((19, 3)\), and \((3, 3)\). These represent the potential solutions for ad placements.

Inequalities

Inequalities are mathematical expressions that define the constraints of a linear programming problem. They show the relationship between variables in terms of greater than, less than, or equal to.

In our exercise, inequalities are used to express the minimum requirements for advertisement reach and quantity:

• \(650,000x + 150,000y \geq 3,000,000\): This inequality ensures the total readership goal.
• \(x \geq 3\) and \(y \geq 3\): These inequalities ensure a minimum number of ads in each magazine.

These inequalities define the feasible region on a coordinate plane by marking out areas where the requirements are fulfilled. By understanding and graphing these inequalities, students can visually interpret which combinations of advertisement placements are possible within the given limits.