Q.Refer to Exercise 14. How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit?

Exercise 14 (sweaters). Type A costs ₹360 and Type B ₹120 to make; at most $300$ sweaters and at most ₹72000 a day; Type B cannot exceed Type A by more than $100$. Profit ₹200 (A), ₹120 (B). With $x$ = Type A, $y$ = Type B:

i.e. $3x+y\le600$, $x+y\le300$, $y-x\le100$.

Corners and $Z$:

• $(0,0):0$
• $(200,0):40000$ …

The referenced problem (Exercise 14)

A company makes Type A sweaters (cost ₹360) and Type B (cost ₹120). It can make at most $300$ sweaters and spend at most ₹72000 a day, and the number of Type B cannot exceed the number of Type A by more than $100$. Profit is ₹200 on A and ₹120 on B. Let $x$ = Type A, $y$ = Type B.

which simplify (divide the first by $120$) to

Step 1 — Corner points

• $(0,0)$.
• $3x+y=600$ with $y=0\Rightarrow(200,0)$ (check $x+y=200\le300$).
• $3x+y=600$ and $x+y=300$: subtract $\Rightarrow2x=300\Rightarrow x=150,\,y=150\Rightarrow(150,150)$ (check $y-x=0\le100$).
• $x+y=300$ and $y-x=100$: add $\Rightarrow2y=400\Rightarrow y=200,\,x=100\Rightarrow(100,200)$ (check $3x+y=500\le600$). …

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: Not simplifying (or wrongly dividing) the budget constraint.

Why it's wrong: leaving $360x + 120y \le 72000$ un-reduced invites arithmetic slips; dividing by the wrong number distorts the line. Correct approach: divide the whole inequality by 120 to get the clean $3x + y \le 600$, which has the same feasible region.

Mistake 2: Getting the comparative constraint backwards.

Why it's wrong: "B may exceed A by at most 100" is $y - x \le 100$; writing $y - x \ge 100$ or $x - y \le 100$ flips the region. Correct approach: $y - x \le 100$. …