Q.Refer to Exercise 13. Solve the linear programming problem and determine the maximum profit to the manufacturer.

Exercise 13 (nickel–chromium products). Each unit of A uses $3$ g nickel and $1$ g chromium; each unit of B uses $1$ g nickel and $2$ g chromium. Available: $9$ g nickel, $8$ g chromium. Profit ₹40 on A, ₹50 on B. With $x$ = units of A, $y$ = units of B:

Corners and $Z$: …

The referenced problem (Exercise 13)

A company makes products A and B. Each A needs $3$ g nickel and $1$ g chromium; each B needs $1$ g nickel and $2$ g chromium. The firm has $9$ g nickel and $8$ g chromium, and earns ₹40 per unit of A and ₹50 per unit of B. Let $x$ = units of A, $y$ = units of B.

Step 1 — Corner points

Boundary lines: $3x+y=9$ (intercepts $(3,0),(0,9)$) and $x+2y=8$ (intercepts $(8,0),(0,4)$).

• $(0,0)$.
• $3x+y=9$ with $y=0\Rightarrow(3,0)$ (check $x+2y=3\le8$).
• $x+2y=8$ with $x=0\Rightarrow(0,4)$ (check $3x+y=4\le9$). …

Method: Graphical (Corner-Point) Method for a Two-Variable LPP

Once a problem is written as "optimise $Z = ax + by$ subject to linear inequalities in $x$ and $y$," this is the standard CBSE technique for finding the optimum.

Steps

Step 1: Draw every constraint as a line.

Replace each inequality by an equation and plot the line from its axis intercepts. Include $x = 0$ and $y = 0$.

Step 2: Shade the correct half-plane.

Test the origin $(0,0)$ in each inequality: if it is satisfied, keep the origin's side; if not, take the other side. Non-negativity confines you to the first quadrant.

Step 3: Identify the feasible region.

It is the single region where all the shaded half-planes overlap.

Step 4: Find the corner points.

Each vertex is the intersection of two boundary lines, found by solving those two equations together. Keep a candidate only if it satisfies every other constraint -- an intersection that breaks a third inequality is not a corner of the region.

Step 5: Evaluate $Z$ at each corner. …

Common Mistakes

Mistake 1: Mixing up which product consumes which material.

Why it's wrong: swapping the coefficients (e.g. writing $x + 2y \le 9$ for nickel) builds the wrong feasible region. Correct approach: match each material to its own row -- nickel $3x + y \le 9$, chromium $x + 2y \le 8$.

Mistake 2: Checking only the axis intercepts. …