6.5/6.6 Optimization Problems II/III: Exploring Solutions and Linear Programming

Pre-Lesson Activity

1. Melody works two jobs. As a waitress, she earns about $12.60 an hour. As a tutor, she earns 20 dollars an hour.

a. Write an objective function that represents Melody’s total income.

b. How much would Melody earn if she works at the restaurant for 15 hours this week and tutors for 6 hours?

c. Would she earn more or less if she works at the restaurant 10 hours only and tutors for 5 hours?

d. Write an objective function that represents Melody’s total hours of work.

e. How can we maximize Melody’s income (assuming a maximum number of hours at work) OR minimize the number of hours she work (assuming a minimum income)?

Example 1

A fast food concession stand sells hotdogs and hamburgers.

Statement Equation Restrictions

Daily sales can be as high as 300 hamburgers and hot dogs combined

At least 1 hamburger must be sold for every 2 hotdogs

Hamburgers and hotdogs are $5.25 and $2.75 respectively

How much would the fast food concession earn if they sold 50 hamburgers and 50 hot dogs?

What is the maximum amount they can earn, given the constraints?

Finding the maximum/minimum of the objective function

The maximum or minimum combination (x number of hamburgers, y number of hot dogs) is usually at or close to the vertices of your solutions region.

What are the vertices of our solutions region above? What kind of a solution do they give?

Co-ordinate on Vertices Solution

Example 2

(Page 334 Foundations of Mathematics 11 by Nelson, 2011 edition)