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Chapter 4: Problem 8

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). $$ y \geq 3 x $$

Short Answer

Expert verified
The region corresponding to the inequality \(y \geq 3x\) is the entire area above the line \(y = 3x\), including the dashed line itself. This region is unbounded, and there are no corner points.

Step by step solution

01

Sketch the boundary line on the coordinate plane.

First, we will sketch the boundary of the inequality. The equation y = 3x represents the boundary line. It passes through the origin (0, 0) and has a slope of 3, which means for every unit increase in x, y will increase by 3 units. Draw the line and make a dashed line since the inequality is "greater than or equal to."
02

Identify the region satisfying the inequality.

Next, we need to find which region corresponds to the given inequality y >= 3x. Since it is "greater than or equal to," we are looking for the region above the line. This region represents all coordinates (x, y) such that y is greater than or equal to 3x.
03

Determine if the region is bounded or unbounded.

In this case, there are no other constraints on the x and y values, so there are no boundaries beyond the line y = 3x. Thus, the region is unbounded.
04

Find the coordinates of all corner points (if any).

Since the region is unbounded, there are no corner points in this problem. The region extends infinitely in the direction above the line y = 3x. In conclusion, the region corresponding to the inequality y >= 3x is the entire area above the line y = 3x, including the dashed line itself. This region is unbounded, and there are no corner points.

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Most popular questions from this chapter

$$ \begin{array}{ll} \text { Minimize } & c=s+t+2 u \\ \text { subject to } & s+2 t+2 u \geq 60 \\ & 2 s+t+3 u \geq 60 \\ & s+3 t+6 u \geq 60 \\ & s \geq 0, t \geq 0, u \geq 0 . \end{array} $$

$$ \begin{array}{ll} \text { Minimize } & c=0.4 s+0.1 t \\ \text { subject to } & 30 s+20 t \geq 600 \\ & 0.1 s+0.4 t \geq 4 \\ & 0.2 s+0.3 t \geq 4.5 \\ & s \geq 0, t \geq 0 \end{array} $$

$$ \begin{aligned} \text { Minimize } & c=s+t+u \\ \text { subject to } & 3 s+2 t+u \geq 60 \\ & 2 s+t+3 u \geq 60 \\ & s+3 t+2 u \geq 60 \\ & s \geq 0, t \geq 0, u \geq 0 . \end{aligned} $$

Your salami manufacturing plant can order up to 1,000 pounds of pork and 2,400 pounds of beef per day for use in manufacturing its two specialties: "Count Dracula Salami" and "Frankenstein Sausage." Production of the Count Dracula variety requires 1 pound of pork and 3 pounds of beef for each salami, while the Frankenstein variety requires 2 pounds of pork and 2 pounds of beef for every sausage. In view of your heavy investment in advertising Count Dracula Salami, you have decided that at least onethird of the total production should be Count Dracula. On the other hand, due to the health-conscious consumer climate. your Frankenstein Sausage (sold as having less beef) is earning your company a profit of \(\$ 3\) per sausage, while sales of the Count Dracula variety are down and it is earning your company only \(\$ 1\) per salami. Given these restrictions, how many of each kind of sausage should you produce to maximize profits, and what is the maximum possible profit? HINT [See Example 3.]

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