/*! 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 27 Solve each linear programming pr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 27

Solve each linear programming problem by the method of corners. Find the maximum and minimum of \(P=10 x+12 y\) subject to $$ \begin{aligned} 5 x+2 y & \geq 63 \\ x+y & \geq 18 \\ 3 x+2 y & \leq 51 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Short Answer

Expert verified
The corner points of the feasible region are \((0, 31.5)\), \((9, 9)\), and \((17, 0)\). Evaluating the objective function \(P = 10x + 12y\) at these corner points, we have: 1. At \((0, 31.5)\), \(P = 10(0) + 12(31.5) = 378\) 2. At \((9, 9)\), \(P = 10(9) + 12(9) = 198\) 3. At \((17, 0)\), \(P = 10(17) + 12(0) = 170\) The maximum value of \(P\) is \(P_{max} = 378\) at the corner point \((0, 31.5)\), and the minimum value of \(P\) is \(P_{min} = 170\) at the corner point \((17, 0)\).

Step by step solution

01

Graph the constraints

First, let's graph the constraints to visualize the feasible region. The constraints are given by: 1. \(5x + 2y \geq 63\) 2. \(x + y \geq 18\) 3. \(3x + 2y \leq 51\) 4. \(x \geq 0\) 5. \(y \geq 0\) Graph these inequalities on the coordinate plane. The feasible region is the area in which all the constraints are satisfied simultaneously.
02

Identify the corner points of the feasible region

Inspect the graph and find the points where the boundary lines intersect each other. These points are the corner points of the feasible region. Note down the exact coordinates of these intersecting points, as they will be used to evaluate the objective function.
03

Evaluate the objective function at the corner points

Evaluate the objective function, \(P = 10x + 12y\), at each corner point identified in the previous step. List the values of the objective function for each point.
04

Determine the maximum and minimum

Compare the values calculated in the previous step, and select the highest and lowest values. The highest value corresponds to the maximum, and the lowest value corresponds to the minimum of the objective function \(P\). Identify the corner points at which these maximum and minimum values are achieved.

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

Find the graphical solution of each inequality. $$ 5 x-3 y \geq 15 $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Suppose you are given the following linear programming problem: Maximize \(P=a x+b y\), where \(a>0\) and \(b>0\), on the feasible set \(S\) shown in the accompanying figure. Explain, without using Theorem 1 , why the optimal solution of the linear programming problem cannot occur at the point \(Q\) unless the problem has infinitely many solutions lying along the line segment joining the vertices \(A\) and \(B\). Hint: Let \(A\left(x_{1}, y_{1}\right)\) and \(B\left(x_{2}, y_{2}\right)\). Then \(Q(\bar{x}, \bar{y})\), where \(\bar{x}=x_{1}+\) \(\left(x_{2}-x_{1}\right) t\) and \(\bar{y}=y_{1}+\left(y_{2}-y_{1}\right) t\) with \(0

Solve each linear programming problem by the method of corners. $$ \begin{array}{ll} \text { Maximize } & P=x+2 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ x & \geq 0, y \geq 0 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{lr} \text { Maximize } P=x+3 y \\ \text { subject to } \quad 2 x+y \leq 6 \\ x+y \leq 4 \\ x \leq 1 \\ x \geq 0, y \geq 0 \end{array} $$

ADVERTISING As part of a campaign to promote its annual clearance sale, the Excelsior Company decided to buy television advertising time on Station KAOS. Excelsior's advertising budget is \(\$ 102,000\). Morning time costs \(\$ 3000 /\) minute, afternoon time costs \(\$ 1000 /\) minute, and evening (prime) time costs \(\$ 12,000 /\) minute. Because of previous commitments, KAOS cannot offer Excelsior more than 6 min of prime time or more than a total of 25 min of advertising time over the 2 weeks in which the commercials are to be run. KAOS estimates that morning commercials are seen by 200,000 people, afternoon commercials are seen by 100,000 people, and evening com- mercials are seen by 600,000 people. How much morning, afternoon, and evening advertising time should Excelsior buy in order to maximize exposure of its commercials?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.