A bakery makes cakes and pastries. The profit is ₹40 per cake and ₹30 per pastry. The resources are 2 hours of baking, 4 hours of packaging and 10 units of flour. Each cake needs 0.5 h baking, 0.2 h packaging, 3 units flour. Each pastry needs 0.3 h baking, 0.4 h packaging, 2 units flour. Using the graphical method, what is the maximum profit and how many cakes and pastries should be produced?

Step-by-step solution

1. Form objective: $Z=40x_1+30x_2$ where $x_1$=cakes, $x_2$=pastries
2. Write constraints: 1. Baking: $0.5x_1+0.3x_{2}le2$\n2. Packaging: $0.2x_1+0.4x_{2}le4$\n3. Flour: $3x_1+2x_{2}le10$\nAll $x_1,x_{2}ge0$
3. Convert inequalities to equalities with slack variables and sketch feasible region in $x_1$‑$x_2$ plane
4. Determine intercepts of each line: - Baking: $x_1=4$, $x_2=\frac{20}{3}$ - Packaging: $x_1=20$, $x_2=10$ - Flour: $x_1=\frac{10}{3}$, $x_2=5$
5. Identify corner points: (0,0), (0,5), (0,10), (4,0), (20,0)
6. Evaluate $Z$ at each corner: $(4,0):160$, $(0,5):150$, $(0,10):300$, $(20,0):800$, $(10,0):400$
7. Largest $Z$ is ₹960 at $(10,0)$
8. Hence optimum: 10 cakes, 0 pastries, profit ₹960

Concept tested: Graphical solution of 2‑variable LP

Published 2 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked

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