Understanding Linear Inequalities and the Fundamental Theorem
School
University of Toronto**We aren't endorsed by this school
Course
MATA 32
Subject
Mathematics
Date
Dec 12, 2024
Pages
1
Uploaded by SuperWillpowerGorilla55
MATA33 - Summer 2023Assignment 1Due:Not to be handed in.•This assignment is based on the material we covered in class during Week 1 (May 8 - 12).•This material will be discussed during the Week 2 tutorials (May 15 - 19).Brief description of material covered during Week 1:•Linear inequalities in two variables & graphical solutions.•Objective function, feasible set/region, nonempty, standard, bounded, corner points.•The Fundamental Theorem of Linear Programming (FTLP):–A linear function defined on a standard feasible region has both a maximum and aminimum value. Moreover, each of these extreme values occurs at a corner point.Notes:•We shade the “anti-region”, i.e., the portion of the plane that isnotthe feasible region.We use solid lines for boundary lines that are included in the feasible region and dashedlines that are not included.•Definition not in the textbook: Astandardfeasible region is a bounded feasible regionwhich includes all corners, edges, and a nonempty interior.Problems from the textbook:•Section 7.1: #1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.•Section 7.2: #1, 3.Extra problems:1. LetRrepresent the feasible region for the constraintsx≥ -2,x-y≤1 and-1≤y≤1.(a) Give a labeled diagram of the feasible regionR.(b) Determine the maximum value ofZ=-2x+yonR(if it exists).2. A store sells two types of cameras. It must sell at least 50 cameras per week, and in orderto satisfy distribution requirements, it must sell at least twice as many of type I as type II.Letxbe the number of type I cameras andybe the number of type II that the store sells ina week. Write a system of inequalities to describe the situation. Find the region describedby the system of linear inequalities.
