Math 1010 Project 2 Linearprogramming

Name: Miranda Paige
Course: Math1010 Section: 408
Optimizing an Advertising Campaign

Math 1010 Intermediate Algebra Project
Background Information:
Linear Programming is a technique used for optimization of a real-world situation. Examples of
optimization include maximizing the number of items that can be manufactured or minimizing the cost
of production. The equation that represents the quantity to be optimized is called the objective
function, since the objective of the process is to optimize the value. In this project the objective is to
maximize the number of people who will be reached by an advertising campaign.
The objective is subject to limitations or constraints that are represented by inequalities. Limitations on
the number of items that can be produced, the number of hours that workers are available, and the
amount of land a farmer has for crops are examples of constraints that can be represented using
inequalities. Broadcasting an infinite number of advertisements is not a realistic goal. In this project
one of the constraints will be based on an advertising budget.
Graphing the system of inequalities based on the constraints provides a visual representation of the
possible solutions to the problem. If the graph is a closed region, it can be shown that the values that
optimize the objective function will occur at one of the "corners" of the region.
The Problem:
In this project you will solve the following situation:
A local business plans on advertising their new product by purchasing advertisements on the radio and
on TV. The business plans to purchase at least 60 total ads and they want to have at most twice as many
radio ads as TV ads. Radio ads cost $20 each and TV ads cost $80 each. The advertising budget is
$4320. It is estimated that each radio ad will be heard by 1750 listeners and each TV ad will be seen by
2500 people. How many of each type of ad should be purchased to maximize the number of people
who will be reached by the advertisements? This report should be submitted with the assumption that it
is a report that will be presented to your employer. Neatness and organization will be a part of the
grade. Anything I have to search too hard for will be considered upgradeable. If you have doubts about
your work, share this with your own employer or a person in authority for feedback.
Submission:
1. To submit this assignment you will post it in your ePortfolio. Give the file a descriptive name,
not just Project 2. After you post your assignment, send me an email in ALEKS with the web
address of your ePortfolio home page.
2. Email your project to me in a Word file to: theprof@professormathguy.com
If you do not do both of these things, your project will not be graded.
Modeling the Problem:

Let X be the number of radio ads that are purchased and Y be the number of TV ads. Please use the
Equation editor to insert the desired inequalities and function. You will find the editor under the Insert
tab.
1. Write down a linear inequality for the total number of desired ads.
x+y60
2. Write down a linear inequality for the cost of the ads.
20x+80y4320
3. Recall that the business wants at most twice as many radio ads as TV ads. Write down a linear
inequality that expresses this fact.
y2x
4. There are two more constraints that must be met. These relate to the fact that there cannot be s
negative numbers of advertisements. Write the two inequalities that model these constraints:
x0
y0
5. Next, write down the function for the number of people that will be exposed to the
advertisements. This is the Objective Function for the problem.
1750x+2500y
You now have five linear inequalities and an objective function. These together describe the situation.
This combined set of inequalities and objective function make up what is known mathematically as a
linear programming problem. Write all of the inequalities and the objective function together below.
This is typically written as a list of constraints, with the objective function last.
x0
y0
y2x
x+y60
20x+80y4320
P=1750x+2500y
6. On your own paper perform the computations necessary to construct the graph. When you have
completed the computations, scan them and insert them after this page in to page(s) of this
document. If you need more than one page to complete your calculations, use only one side of a
clean sheet of paper for each page and insert each page you create into its own document page.
The overlapping shaded portion of the graph you will create is called the feasible region. Any
(x,y) point in the feasible region corresponds to a possible number of radio and TV ads that will
meet the requirements and constraints of the problem. However, the value that will maximize
the number of people exposed to the ads will occur at one of the three vertices or corners of the
feasible region. Find the coordinates of these corners by solving the appropriate systems of
Linear equations. Be sure to keep your work (done by hand) EXTREMELY neat and organized.
Remember continued employment will depend on the quality of the work you do.
7. After you have completed step 6, list the points necessary to construct the graph. Three of these
points will be the vertices of the feasible region. Using the equation editor, enter the three
vertices here.
(20,40), (24,48), (8, 52)
8. To solve this problem, you will need to graph the intersection of all five inequalities on one
common XY plane. Do this using one of the graphing programs suggested in Project 1. I
strongly suggest that you use the Graph for Windows program found at:
http://www.padowan.dk/graph/
Of the four programs, this is the most intuitive and easiest to display all of the qualities I want to
see in the graph. Have the bottom left of the graph be the origin, with the horizontal axis
representing X and the vertical axis representing Y. Label the axes with what they represent (not
just x and y.) and label your lines as you graph them. Also, label each of the vertices.
Remember, you are graphing inequalities and they should be appropriately shaded. Make sure
that you set your parameters for the graph so it mainly shows the positive quadrant only. You
will need to specify the minimum tics at about -3 for both x and y for the axes to display.
Optimization to maximize the viewing of a combination of radio and TV adds
9. To find which number of radio and TV ads will maximize the number of people who are exposed to the
business advertisements, evaluate the objective function P for each of the vertices you found. Show your
work. Do these calculations by hand and insert them on this page. These calculations should be relatively
simple and take up little space. You will need to crop your scan so that it fits in the space below
10. Write a complete sentence describing how many of each type of advertisement should be purchased and
what is the maximum number of people who will be exposed to the ad.
The maximum number of people exposed to the ads is 162,000 people, if 24 radio ads and 48 TV ads are
purchased.
11. Reflective Writing.

Did this project change the way you think about how math can be applied to the real world? Write one
or two paragraphs stating what ideas changed and why. If this project did not change the way you think,
write how this project gave further evidence to support your existing opinion about applying math. Be
specific.
This project showed me that there are equations that are relevant to real world problems. It
showed me that I could more easily solve problems by making an equation and solve it by myself in my
own life. Complex math equations are more common in the world that I imagined, and I have no doubt
that the business and marketing fields use linear equations very often to be able to maximize or minimize
profits. It can also help managers figure out pay roll and lunch break time periods to be able to get the
results they want. It can help the food industries find out how many orders of food they need and how to
minimize waste or unused food products.
This type of problem could be more challenging to use in my life, because I dont have that
many opportunities that would entitle a need for linear programming. But as I will progress throughout
school and career opportunities, I would be able to at least know how to solve certain problems like this
one and it has made me more of a resourceful person. Since I am going into the field of psychology, I
need to know how to problem solve really well. My new knowledge of how to apply mathematical
equations to real life problems could benefit me, and my future co-workers and patients, greatly.

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Math 1010 Project 2 Linearprogramming

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Name: Miranda Paige

Course: Math1010 Section: 408

Optimizing an Advertising Campaign

Modeling the Problem:

11. Reflective Writing.

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