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Chapter 4: Problem 19

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} &20 x+10 y \leq 100 \\ &10 x+20 y \leq 100 \\ &10 x+10 y \leq 60 \\ &x \geq 0, y \geq 0 \end{aligned} $$

Short Answer

Expert verified
The region defined by the given inequalities is a bounded triangle with corner points at \((0,5)\), \((2,4)\), and \((4,0)\).

Step by step solution

01

Graph Each Inequality

To graph each inequality presented in the system, first treat each inequality as an equation to find the corresponding line. Then, we determine where the inequality is true (above or below the line). $$ \begin{aligned} &20x + 10y = 100 \Rightarrow y = -2x + 10\\ &10x + 20y = 100 \Rightarrow y = -\frac{1}{2}x + 5\\ &10x + 10y = 60 \Rightarrow y = -x + 6 \end{aligned} $$ While drawing the inequalities, remember to include the nonnegative constraints: \(x \geq 0\) and \(y \geq 0\). Shade the region that satisfies all inequalities at once.
02

Bounded or Unbounded Region

Look at the graph drawn in Step 1 and determine if the intersection region has a bounded or unbounded area. If the region has finite area, it is bounded, otherwise, it is unbounded.
03

Find Corner Points

There may be intersection points among the inequalities which can be considered as corner points of the region defined by the inequality system. To find these corner points, first, identify the intersection points of each inequality. Then, verify if these points lie within the defined region. 1. Intersection of \(20x + 10y \leq 100\) and \(10x + 20y \leq 100\): $$ \begin{cases} 20x + 10y = 100\\ 10x + 20y = 100 \end{cases} $$ 2. Intersection of \(20x + 10y \leq 100\) and \(10x + 10y \leq 60\): $$ \begin{cases} 20x + 10y = 100\\ 10x + 10y = 60 \end{cases} $$ 3. Intersection of \(10x + 20y \leq 100\) and \(10x + 10y \leq 60\): $$ \begin{cases} 10x + 20y = 100\\ 10x + 10y = 60 \end{cases} $$ Note that there might be other corner points due to nonnegative constraints, such as the intersections of lines with the x-axis and y-axis. After finding the intersection points, make sure they satisfy all the inequalities, including the nonnegative constraints. The points that meet all these conditions are the corner points of the defined region.

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Key Concepts

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

Linear Programming
Linear programming (LP) is a mathematical technique used to optimize a linear objective function, subject to a set of linear inequalities or equations, known as constraints. This method is widely employed in various fields such as economics, business, engineering, and military applications to find the best possible solution among a range of feasible solutions.

For example, a company might use linear programming to maximize profits or minimize costs while adhering to production constraints. The process involves graphing the system of inequalities to form a feasible region, where the optimal solution can be found at one of the corner points of this region. LP problems can have either a single optimum solution, multiple solutions, or no solution at all, often depending on the nature of the constraints and the objective function.
Bounded Region
When graphing systems of linear inequalities, one fundamental concept is identifying whether the feasible area is a bounded or unbounded region. A bounded region is an enclosed area on the coordinate plane where all points that satisfy the inequalities are within a finite space. In contrast, an unbounded region extends indefinitely in one or more directions.

Identifying boundedness is crucial in linear programming since if the feasible region is bounded, the optimal solution must lie within the confines of this region. If it is unbounded, it suggests that the objective function may increase or decrease indefinitely, and extra care must be taken to check for the existence of an optimal solution. The graphical method of solving LP problems typically displays a bounded region as a polygon formed by intersecting lines or half-planes, as prescribed by the system of inequalities.
Corner Points
The importance of corner points, also known as vertices, in the context of linear programming cannot be overstated. These are the critical points where the edges of the feasible region intersect. According to the Fundamental Theorem of Linear Programming, if there is an optimal solution to the LP problem, it will occur at one of the corner points.

In practice, to find these points, we solve the equations that represent the borders of the feasible region where inequalities intersect. For example, if two of our constraints are equations of straight lines, the corner point is where these two lines intersect. The concept pairs with the graphical solution method, where you analyze these points to locate the optimal value of the objective function.

It is crucial to check each corner point against all the inequalities to ensure it is indeed part of the feasible region. Any point that does not satisfy all the inequalities does not belong to the feasible solution set and hence is not considered a valid corner point.

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Most popular questions from this chapter

Agriculture \(^{29}\) Your small farm encompasses 100 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: \(\$ 5\) for tomatoes, \(\$ 4\) for lettuce, and \(\$ 2\) for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of \(\$ 2,000\) for each acre of tomatoes, \(\$ 1,500\) for each acre of lettuce and \(\$ 500\) for each acre of carrots. You would like to spend at least \(\$ 400\) on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 500 hours per week. How many acres of each crop should you plant to maximize total profits? In this event, will you be using all 100 acres of your farm? HINT [See Example 3.]

$$ \begin{array}{ll} \text { Maximize } & p=2 x+y \\ \text { subject to } & x+2 y \leq 6 \\ & -x+y \leq 2 \\ & x \geq 0, y \geq 0 . \end{array} $$

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. You make a profit of \(\$ 3\) on each quart of Creamy Vanilla and \(\$ 2\) on each quart of Continental Mocha. How many quarts of each flavor should you make in order to earn the largest profit?

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. \({ }^{11}\) If the cereal costs \(30 \phi\) per serving and the dessert costs 50 per serving, and you want to provide your child with at least 140 calories and at least 32 grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted.)

$$ \begin{array}{cc} \text { Minimize } & c=2 s+t+3 u \\ \text { subject to } & s+t+u \geq 100 \\ & 2 s+t \quad \geq 50 \\ & s \geq 0, t \geq 0, u \geq 0 . \end{array} $$
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