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Background Information:
Linear programming is used to identify conditions that maximize profit or minimize cost. Today we will investigate how much profit a furniture Company will make in one week given a limited amount of resources. Using LEGOS( to model building tables and chairs, you will use linear programming to relate mathematics to the real world by maximizing furniture profits in preparation of North Carolinas famous, semi-annual Furniture Market.
Each group will need a weekly allotment of materials:
A total of 6 4 ( 2 toy LEGOS to represent table tops.
A total of 10 1 ( 1 toy LEGOS to represent chair backs.
A total of 16 2 ( 1 toy LEGOS to represent bases needed for both tabletops and chair backs.
Each group member will need:
A TI-83 Plus or TI-84 Plus graphing calculator with Inequality Graphing APPlication
Investigation:
1. A chair can be built using 1 chair back (1 ( 1 LEGO) and 1 base (2 ( 1 LEGO).
A table can be built using 1 tabletop (4 ( 2 LEGO) and 2 bases.
Take an inventory and confirm that you have your weekly allotment of tabletops, chair backs, and bases to build the furniture.
2. The furniture company makes a profit of $30 for every table it produces and $10 for every chair it produces. Let x represent the number of tables produced in a week and let y represent the number of chairs produced in a week. Using algebra, write an algebraic expression to represent a profit function for the expected total amount of weekly profit.
f(x, y) = ____________________________
3. If your goal is to maximize profit, begin building possible combinations of tables and chairs. Complete the numerical data table with possible combinations of tables and chairs along with their respective profit totals calculated using the above profit function.
(Tables, Chairs)
(x, y)Profit Function
Total Profit
f(x, y)
4. Because of the limited amount of weekly resources in chair backs, tabletops, and bases, use algebra to develop a system of linear inequalities to represent these constraints.
Tabletop constraint: ____________________
Chair back constraint: ____________________
Bases constraint: ____________________
Are there any non-negativity constraints? ____________________
5. You will now use your graphing calculator to obtain a graphical picture of your constraints.
Press APPS and begin the Inequality Graphing APPlication on your graphing calculator.
Put all constrains in the x= and ( menus. Dont forget your non-negativity constraints. Press ( and the appropriate soft key ((, (, (, (, or () to obtain the appropriate inequality symbol needed for your constraint.
Adjust your ( accordingly to obtain a complete graph of the feasible region that represents your constraints.
Press (; (; Shades (soft key ( or (); and 1:Ineq Intersection. The calculator will produce the feasible region. It should be in the shape of a polygon.
6. It is now time to walk the polygonal feasible region collecting corner points as you go. Begin this journey by pressing ( and POI-TRACE (soft key ( or (). To automatically store the corner point (x, y) in a numerical table, press (. The calculator will automatically store this point in the statistical editor as name lists (LINEQX and LINEQY). Assign a formula to L1 that will calculate profits using your original profit function, L1 = 30 ( LINEQX + 10 ( LINEQY
Summarize your results:
1. When building tables and chairs using toy Legos(, what combination of tables and chairs built will produce a maximum profit? What is the maximum profit?
______________________________________________________________________
2. Why did the bases constraint involve both variables, x and y?
_____________________________________________________________________
3. When developing algebraic inequalities to represent the constraints for the limited resources, why should non-negativity be considered?
_____________________________________________________________________
4. When developing the polygonal feasible region, why were there five corner points?
_____________________________________________________________________
5. Was there any correlation between the numerical table developed by hand and the table the graphing calculator developed using corner points? If so, what was that correlation?
_____________________________________________________________________
6. If you were the Chief Executive Officer of a furniture company that produced similar tables and chairs and you wanted to invest more money into one of the three limited resources, which resource would you choose and why?
_____________________________________________________________________
More Independent Practice:
During the summer break, Nathan works as many as 35 hours per week. On Saturdays he spends between two and six hours delivering furniture. On weekdays he can work between 10 and 40 hours at the recreation center as a counselor for the childrens day camp. Delivering furniture pays $10 per hour while the summer counselors job pays $6.25 per hour. How many hours should Nathan work at each job to produce the most income during his summer break?
Algebraic Representation:
1. If x represents delivering furniture and y represents working as a counselor, develop an algebraic expression that represents Nathans maximum summer earnings as a function of hours worked.
2. Develop a system of linear inequalities that represents the constraints (or limitations) to the number of hours Nathan can work. Dont forget non-negativity!
Graphical Representation:
3. Using the Inequality Graphing APPlication, create a visual picture of the constraints.
Numerical Representation:
4. Walk the polygonal feasible region collecting corner points on your journey. Store these points in the statistical editor.
5. Confirm the collection of corner points from the feasible region in the statistical editor. They should be stored in (LINEQX, LINEQY).
6. Assign a formula to L1 that will calculate the maximum amount of income during Nathans summer.
Summarize Your Results Complete the following table:
(Furniture, Counselor)
(x, y)Maximum Earnings Function
Total Earnings
f(x, y)
7. Does the table above agree with the table of earnings from the statistical editor?
8. How many hours at each job should Nathan work to maximize his summer earnings?
9. If Nathan could work more than 35 hours per week, at which job should he work more hours?
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