Linear Optimization Guided Worksheet
Finite Mathematics // Linear Programming
Linear Optimization with Graphing
Name:
Date: Period:
The Graphing Calculator LP Protocol
Corner Point Method
1. Model & Isolate Write constraints and isolate \(y\) for calculator entry: \(Y_1 = \dots, Y_2 = \dots\)
2. Graph & Window Set \([0, X_{\max}]\) and \([0, Y_{\max}]\). Graph lines and identify shaded region.
3. Calculate Intersect Press 2nd TRACE 5 (Intersect) to find each boundary vertex.
4. Evaluate Vertices Evaluate objective formula at every corner point to find max profit or min cost.
Model Problem 1: Apex Electronics Product Mix Guided Walkthrough
Apex manufactures Studio Monitors (\(x\)) and Stage Speakers (\(y\)). Profit: \(P = 50x + 40y\).
Assembly Constraint: \(2x + y \le 100 \implies\) \(Y_1 = 100 - 2x\)
Testing Constraint: \(x + 2y \le 80 \implies\) \(Y_2 = (80 - x)/2 = 40 - 0.5x\)
Non-negativity: \(x \ge 0, y \ge 0\)
Calculator Tip: In your graphing calculator, enter \(Y_1\) and \(Y_2\). Press 2nd CALC 5:intersect. Select curve 1, curve 2, and press Enter to find \((40, 20)\).
Pre-Graphed Feasible Region 0 50 100 (x) 40 100 (y) Y1: 2x+y=100 Y2: x+2y=80 (40, 20) (50, 0) (0, 40) (0, 0) Shaded area = Feasible Region
Corner Point Evaluation for \(P = 50x + 40y\):
Vertex \((0, 0)\) \(P = 50(0) + 40(0) = \$0\)
Vertex \((50, 0)\) \(P = 50(50) + 40(0) = \$2,500\)
Vertex \((40, 20)\) ★ MAX \(P = 50(40) + 40(20) = \$2,800\)
Vertex \((0, 40)\) \(P = 50(0) + 40(40) = \$1,600\)
Key Conclusion: Optimal production is 40 monitors and 20 speakers for a maximum daily profit of $2,800.
Page 1 of 4 • Graphing Calculator Linear Optimization
PART II
Business Profit Maximization
Lines Pre-Graphed • Calculator Intersect
Problem 1: Heritage Woodcraft
Profit Model
Heritage Woodcraft produces Executive Desks (\(x\)) and Credenzas (\(y\)). Profit: \(P = 250x + 300y\).
• Hardwood Lumber: \(3x + 2y \le 180 \implies Y_1 = (180 - 3x)/2\)
• Carpentry Labor: \(x + 2y \le 100 \implies Y_2 = (100 - x)/2\)
• Non-negativity: \(x \ge 0, y \ge 0\)
Pre-Graphed Lines & Feasible Region 0 60 100 (x) 50 90 (y) Y1: 3x+2y=180 Y2: x+2y=100 Intersect (?) (60, 0) (0, 50)
a Calculator Input Equations
State the equations entered into \(Y_1\) and \(Y_2\):
b Calculator Intersect Result
Use 2nd TRACE 5 to find the intersection of \(Y_1\) and \(Y_2\):
c Evaluate Objective Function: \(P = 250x + 300y\)
List all 4 corner points visible from the graph above and compute profit:
Corner Point \((x, y)\)
Boundary Description
Profit Expression \(P = 250x + 300y\)
Total Profit ($)
\((0, 0)\)
Origin (No production)
\(250(0) + 300(0)\)
$0
(d) Managerial Decision & Resource Slack:
State the optimal units of desks (\(x\)) and credenzas (\(y\)) to maximize profit. Are any lumber board-feet or labor hours left over? Show work:
Finite Mathematics • Unit 4 Page 2 of 4 • Business Profit Maximization
PART III
Linear Cost Minimization
Lines Pre-Graphed • \(\ge\) Constraints
Problem 2: NutriFarm Feed Blend
Minimization Model
NutriFarm mixes Grain Alpha (\(x\) kg) at $0.50/kg and Grain Beta (\(y\) kg) at $0.80/kg. Objective: Minimize \(C = 0.50x + 0.80y\).
• Protein Requirement: \(2x + y \ge 14 \implies Y_1 = 14 - 2x\)
• Mineral Requirement: \(x + 2y \ge 16 \implies Y_2 = (16 - x)/2\)
• Non-negativity: \(x \ge 0, y \ge 0\)
Pre-Graphed Minimization Region 0 7 16 (x) 8 14 (y) Y1: 2x+y=14 Y2: x+2y=16 (0, 14) Intersect (?) (16, 0)
a Calculator Setup
State isolated formulas for \(Y_1\) and \(Y_2\):
b Calculator Intersect Verification
Record the intersection coordinate found with 2nd TRACE 5:
c Evaluate Cost Function: \(C = 0.50x + 0.80y\)
Evaluate the 3 feasible boundary vertices shown on the graph to find the minimum cost:
Feasible Corner Point \((x, y)\)
Cost Calculation \(C = 0.50x + 0.80y\)
Total Blend Cost ($)
(d) Nutritional Procurement Recommendation:
State the exact kilograms of Grain Alpha and Grain Beta to purchase per batch, and state the minimum batch cost:
Finite Mathematics • Unit 4 Page 3 of 4 • Linear Cost Minimization
PART IV
Advanced Feasibility & Troubleshooting
Spotting False Intersections
Problem 3: TechPro Multi-Constraint Mix
3-Constraint Model
TechPro produces Smartwatches (\(x\)) and FitBands (\(y\)). Profit: \(P = 40x + 35y\). Three departmental limits are active:
1. Chipsets: \(x + y \le 12 \implies Y_1 = 12 - x\)
2. Assembly: \(2x + y \le 20 \implies Y_2 = 20 - 2x\)
3. Inspection: \(x + 2y \le 20 \implies Y_3 = (20 - x)/2\)
Pre-Graphed 3-Line Region 0 10 20 (x) 10 20 (y) Y1: x+y=12 False Point (6.7, 6.7) ✘ (8, 4) (4, 8)
a Why is \((6.7, 6.7)\) a "False Corner Point"?
Test \((6.67, 6.67)\) in Constraint 1 (\(x + y \le 12\)). Explain why it cannot be the optimal solution:
b Confirm the True Feasible Vertices
Use your calculator to verify the intersection of \(Y_1\) and \(Y_2\), and \(Y_1\) and \(Y_3\):
c Evaluate Profit \(P = 40x + 35y\) at the True Feasible Vertices
\((0, 0)\)
\((10, 0)\)
\((8, 4)\)
\((4, 8)\)
\((0, 10)\)
Golden Rule of Graphing Calculator Optimization:
Why must you always inspect the shaded feasible region on your calculator before blindly recording intersection points from 2nd TRACE 5?
Finite Mathematics • Unit 4 Page 4 of 4 • Multi-Constraint Troubleshooting
