Linear Programming Project

Name: ______Block: ______Due: ______

LINEAR PROGRAMMING PROJECT

You are the new owner of a music shop in Greenwood. The previous owner fled the city to join the circus as a magician . Your first duty as new owner and store manager is to create an advertising plan based on the budget available. You must figure out how many magazine and TV ads to purchase.

 TV ads cost $600 per airing.

 Magazine ads cost $1200 per issue.

 You total advertising budget is $9,000.

1. If we let x = TV ads and y = magazine ads, write an inequality for our advertising budget.

2. Due to space limitations, the magazine publishers tell us that we are only allowed to purchase up to 6 magazine ads. Write an inequality for this constraint.

3. The television station called to say that we are only allowed to purchase up to 7 TV ads. Write an inequality for this constraint.

4. It is impossible to buy a negative number of TV ads. Write an inequality for this constraint.

5. It is impossible to buy a negative number of magazine ads. Write an inequality for this constraint.

Graph this system of inequalities on the graph paper provided. This will help us determine our region of feasible solutions. Name: ______Block: ___ Date:______

Graph the system of inequalities you found on the previous page. When you’re finished. Clearly outline your region of feasible solutions (where all of the graphs overlap).

Determine the five points where the graphs intersect. These are called the vertices of the region of feasible solutions. Show work if necessary.

1. 2. 3.

4. 5. Name: ______Block: ___ Date:______

Television Ads The following data was collected in past years to try to determine how TV ads affect CD sales. Plot the points and estimate the line of best fit.

Number Increase in 0 0

of TV CD sales 0 1

ads 0 0

5 725 0 0 8

6 900 e l 7

a 4 450 s 0

3 400 D 6

5 750 0 n 0 i 5

3 600 e

2 350 0 e 4

4 575 c 0 n 0 I 3 450 3

5 700 0 2

1 150 0

2 325 0 1

6 950 0 0 1 2 3 4 5 6 7 8 9 10

Number of TV Ads

Use the points (1, 150) and (6, 900) to estimate the line of best fit. 1. Graph the line that goes through both points. 2. Find the slope of the line.

3. Write the equation of the line.

4. What does the slope tell you about how each TV ad affects sales?

For every TV ad, CD sales increased by about ______. Name: ______Block: ___ Date:______affect CDsales.affect datacollected The years following was toads determinepast tohowin try magazine Name: ______Block: ___ Date:______Block:______Name: Usethepoints(1, 100) (8,and 800) toestimate the ofbest line fit. magazine Number ads 4. 3. 2. 1. of 5 2 4 7 3 6 2 9 5 8 4 7 6 8 1 sales? does What slope the abouttellyou howeach magazine ad affects the Write equation the of line. Find slope the line. the of Graphline the goes that through both points. For everyFor magazine salesad, CD increased aboutby ______. Increase in Increase CD sales 560 275 410 640 300 630 150 900 440 800 375 725 590 725 100 Plot the points the theestimate Plot linebest points of and fit.

0 100 200 300 400 500 600 700 800 900 1000

Increase in CD sales MagazineAds 0 12345678910 Number of Magazine Magazine NumberAds of We now have all of the information we need to solve the linear program. 1. Write an objective function for CD sales. (Hint: Think about how each TV and magazine ad affects sales.)

2. Substitute the coordinates of the vertices into the objective function.

(objective function) (x, y) f (x, y) Vertex Point Total Sales

3. What is the maximum and where did it occur?

4. Knowing this information, how many TV and magazine ads should you buy?

CONGRATULATIONS!!! You’re finally FINISHED! Name: ______Block: ___ Date:______

Extra Credit Now that you have found your optimal solution, think about how a change in our allowable advertising values might affect the solution. This is called sensitivity analysis. Your job now is to find out how much a change in one of your constraint values will affect your final profit.

 For example, what if instead of being allowed to buy 7 television commercials, you were only allowed to buy 5. What would be the new optimal solution? (show your work)

 If you were allowed to increase one constraint value in order to increase your profit, which one should you change? In other words, would you rather be allowed to buy one more TV ad or one more magazine ad? Why?

LINEAR PROGRAMMING PROJECT

You are the new owner of a music shop in Greenwood. The previous owner fled the city to join the circus as a magician . Your first duty as new owner and store manager is to create an advertising plan based on the budget available. You must figure out how many magazine and TV ads to purchase.

 TV ads cost $600 per airing.

 Magazine ads cost $1200 per issue.

 You total advertising budget is $9,000.

6. If we let x = TV ads and y = magazine ads, write an inequality for our advertising budget.

7. Due to space limitations, the magazine publishers tell us that we are only allowed to purchase up to 6 magazine ads. Write an inequality for this constraint.

8. The television station called to say that we are only allowed to purchase up to 7 TV ads. Write an inequality for this constraint.

9. It is impossible to buy a negative number of TV ads. Write an inequality for this constraint.

10.It is impossible to buy a negative number of magazine ads. Write an inequality for this constraint.

Graph this system of inequalities on the graph paper provided. This will help us determine our region of feasible solutions.

Name: ______Block: ___ Date:______Graph the system of inequalities you found on the previous page. When you’re finished. Clearly outline your region of feasible solutions (where all of the graphs overlap).

Determine the five points where the graphs intersect. These are called the vertices of the region of feasible solutions. Show work if necessary.

1. (0, 0) 2. (7, 0) 3. (0, 6)

4. (7, 4) 5. (3, 6)

Name: ______Block: ___ Date:______

Television Ads The following data was collected in past years to try to determine how TV ads affect CD sales. Plot the points and estimate the line of best fit.

Number Increase in 0 0

of TV CD sales 0 1

ads 0 0

5 725 0 0 8

6 900 e l 7

4 450 s

3 400 D 6

5 750 0 n 0 i

3 600 e

2 350 a 0 e 4 r

4 575 c 0 n 0 I

3 450 3

5 700 0 0 2

1 150 0

2 325 0 1

6 950 0 0 1 2 3 4 5 6 7 8 9 10

Number of TV Ads

Use the points (1, 150) and (6, 900) to estimate the line of best fit. 5. Graph the line that goes through both points. 6. Find the slope of the line. m = 150

7. Write the equation of the line. y = 150x

8. What does the slope tell you about how each TV ad affects sales? For every TV ad, CD sales increased by about 150 Name: ______Block: ___ Date:______Magazine Ads The following data was collected in past years to try to determine how magazine ads affect CD sales. Plot the points and estimate the line of best fit. Usethepoints(1, 100) (8,and 800) toestimate the ofbest line fit. We now have all of the information we need to solve the linear program. linear needto program. we ofsolve the information nowhave all the We Date:______Block:______Name: magazine Number ads 5. 5. 8. 7. 6. of 5 2 4 7 3 6 2 9 5 8 4 7 6 8 1 For everyFor magazine salesad, CD increased aboutby sales? does What slope the abouttellyou howeach magazine ad affects the Write equation the of line. Find slope the line. the of Graphline the goes that through both points. TV and magazine ad affects sales.)affects ad magazine TV and howThink about each (Hint: CD sales. function objective for an Write Increase in Increase CD sales 560 275 410 640 300 630 150 900 440 800 375 725 590 725 100

0 100 200 300 400 500 600 700 800 900 1000

Increase in CD sales m y 0 12345678910 = = 100 = = 100 x Number of Magazine Magazine NumberAds of 100 6. Substitute the coordinates of the vertices into the objective function.

(objective function) (x, y) f (x, y) Vertex Point Total Sales (0, 0) 150(0) + 100(0) 0

(7, 0) 150(7) + 100(0) 1050

(0, 6) 150(0) + 100(6) 600

(7, 4) 150(7) + 100(4) 1450

(3, 6) 150(3) + 100(6) 1050

7. What is the maximum and where did it occur? Maximum value is $1450 It occurs at (7, 4)

8. Knowing this information, how many TV and magazine ads should you buy?

You should buy 7 TV and 4 magazine ads

CONGRATULATIONS!!! You’re finally FINISHED!