/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Sketch the region that correspon... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

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). $$ \begin{aligned} -3 x+2 y & \leq 5 \\ 3 x-2 y & \geq 6 \\ y & \leq x / 2 \\ x \geq 0, y \geq & 0 \end{aligned} $$

Short Answer

Expert verified
The region corresponding to the given inequalities is bounded and lies in the first quadrant. The corner points of the region are \(P_1=(0,0)\), \(P_2=(2,1)\), and \(P_3=(4,5)\).

Step by step solution

01

Rewrite inequalities as equalities

Rewrite the inequalities as equalities to graph them: $$ \begin{aligned} -3x + 2y &= 5 \\ 3x - 2y &= 6 \\ y &= \frac{x}{2} \end{aligned} $$
02

Sketch the individual lines

To graph the lines, first rewrite equations in slope-intercept form (y = mx + b) where possible, then plot the lines: $$ \begin{aligned} y &= \frac{3}{2}x + \frac{5}{2} \\ y &= \frac{3}{2}x - 3 \\ y &= \frac{x}{2} \end{aligned} $$ Also, note the two conditions \( x \geq 0 \) and \( y \geq 0 \). This means that the region lies in the first quadrant.
03

Identify the region formed by the inequalities

Now that we have the lines plotted, we need to fill in the regions determined by the inequalities. Recall the inequalities and their corresponding lines: $$ \begin{aligned} -3 x+2 y & \leq 5 & (1) \ \ y &= \frac{3}{2}x + \frac{5}{2} \\ 3 x-2 y & \geq 6 & (2) \ \ y &= \frac{3}{2}x - 3 \\ y & \leq x / 2 & (3) \ \ y &= \frac{x}{2} \\ x \geq 0, y \geq & 0 & (4) \end{aligned} $$ (1) Keep the region below or on the line. (2) Keep the region above or on the line. (3) Keep the region below or on the line. (4) Keep the region in the first quadrant.
04

Check if the region is bounded or unbounded

A region is bounded when it's contained within a finite area. Looking at our graph, the region formed by the intersections of the lines and inequalities is enclosed, so the region is bounded.
05

Find the corner points of the region

Corner points are the points of intersection of the boundary lines. We have 3 corner points: 1. Intersection of lines (1), (3), and (4): \( P_1 = (0, 0) \) 2. Intersection of lines (1) and (3): Solving the system of equations, $$ \begin{cases} y = \frac{3}{2}x + \frac{5}{2} \\ y = \frac{x}{2} \end{cases} $$ We find that \( P_2 = (2,1) \). 3. Intersection of lines (1) and (2): Solving the system of equations, $$ \begin{cases} y = \frac{3}{2}x + \frac{5}{2} \\ y = \frac{3}{2}x - 3 \end{cases} $$ We find that \( P_3 = (4,5) \). So, the corner points are \(P_1=(0,0)\), \(P_2=(2,1)\), and \(P_3=(4,5)\).

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 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Give an example of a standard minimization problem whose dual is not a standard maximization problem. How would you go about solving your problem?

$$ P=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 1 & 2 & 0 \\ 0 & 1 & 1 \end{array}\right] $$

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.]

\(\nabla\) Scheduling Because Joe Slim's brother was recently elected to the State Senate, Joe's financial advisement concern, Inside Information Inc., has been doing a booming trade, even though the financial counseling he offers is quite worthless. (None of his seasoned clients pays the slightest attention to his advice.) Slim charges different hourly rates to different categories of individuals: \(\$ 5,000 /\) hour for private citizens, \(\$ 50,000 /\) hour for corporate executives, and \(\$ 10,000 /\) hour for presidents of universities. Due to his taste for leisure, he feels that he can spend no more than 40 hours/week in consultation. On the other hand, Slim feels that it would be best for his intellect were he to devote at least 10 hours of consultation each week to university presidents. However, Slim always feels somewhat uncomfortable dealing with academics, so he would prefer to spend no more than half his consultation time with university presidents. Furthermore, he likes to think of himself as representing the interests of the common citizen, so he wishes to offer at least 2 more hours of his time each week to private citizens than to corporate executives and university presidents combined. Given all these restrictions, how many hours each week should he spend with each type of client in order to maximize his income?

$$ \begin{array}{ll} \text { Minimize } & c=2 s+t \\ \text { subject to } & 3 s+t \geq 30 \\ & s+t \geq 20 \\ & s+3 t \geq 30 \\ & s \geq 0, t \geq 0 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.