The High School Yearbook Staff Is Deciding On Color Pages

The High School Yearbook Staff Is Deciding How Many Color Pages To Inc

The high school yearbook staff is deciding how many color pages to include in the yearbook. In order to provide the appropriate visual appeal, the yearbook staff has decided that the number of black and white pages must be no more than twice the number of color pages. Due to cost constraints, there can be no more than 40 color pages in the yearbook. Use guess and check to determine three different combinations of pages that will work. Write a system of inequalities that models the situation. Solve the system graphically. show that your three different combinations lie inside the solution region. Explain what the intersection region represents.

Paper For Above instruction

The process of designing a high school yearbook involves considerate planning to balance visual appeal with budget constraints. In this context, the decision of how many color and black-and-white pages to include must adhere to specific conditions, which can be modeled mathematically through a system of inequalities. This paper discusses the creation of such a system, solves it graphically, identifies three valid page combinations, and interprets what the intersection of these conditions signifies.

Modeling the Situation with Inequalities

Let \( C \) represent the number of color pages, and \( B \) represent the number of black-and-white pages, both being non-negative integers since negative pages are nonexistent. Based on the problem, the constraints can be formulated as follows:

1. Black-and-White Pages Constraint: The number of black-and-white pages must be no more than twice the number of color pages.

\[

B \leq 2C

\]

2. Color Pages Constraint: There are no more than 40 color pages due to budget constraints.

\[

C \leq 40

\]

3. Non-negativity Constraints:

\[

C \geq 0, \quad B \geq 0

\]

Given that total pages are not explicitly constrained, these inequalities generate a feasible region in the coordinate plane where \( C \) and \( B \) can vary.

Graphical Solution

Graphing these inequalities involves plotting the boundary lines:

- The line \( B = 2C \) serves as the upper boundary for black-and-white pages relative to color pages.

- The line \( C=40 \) caps the maximum number of color pages.

Since \( B \geq 0 \) and \( C \geq 0 \), the feasible region is bounded in the first quadrant, confined below \( B=2C \) and to the left of \( C=40 \).

Identifying Three Valid Page Combinations

Using guess and check within the feasible region:

1. Combination 1: \( C = 10 \), then \( B \leq 20 \); choosing \( B=20 \) satisfies the constraints.

2. Combination 2: \( C=20 \), then \( B \leq 40 \); choosing \( B=40 \), which is the maximum allowed.

3. Combination 3: \( C=5 \), then \( B \leq 10 \); choosing \( B=10 \).

All these pairs satisfy the inequalities, and thus, are valid combinations of pages.

Graphical Representation

Plotting the points:

- \( (C, B) = (10, 20) \)

- \( (20, 40) \)

- \( (5, 10) \)

these points lie inside the feasible region defined by the system. This region is the triangle (or polygon) bounded by the axes, the line \( B=2C \), and the vertical line \( C=40 \).

Interpretation of the Intersection Region

The intersection region signifies all possible combinations of color and black-and-white pages that meet the constraints. Any point within this region corresponds to a feasible yearbook plan. The solutions reflect the permissible allocation of pages that optimize visual variety while adhering to budget and design constraints. By selecting different points, the staff can diversify the yearbook's visual layout, ensuring an appealing balance that complies with all constraints.

Conclusion

This modeling and graphical solution process allows the yearbook staff to visualize the feasible options for page design. The three combinations identified demonstrate the diversity of acceptable choices within the constraints, ensuring flexibility in planning. Understanding the intersection region helps clarify the scope of feasible page combinations, which is essential for effective decision-making in project planning.

References

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  • U.S. Census Bureau. (2022). Budget constraints and project planning. https://www.census.gov
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