Q.A manufacturer produces two types of steel trunks. He has two machines $A$ and $B$. The first type of trunk requires 3 hours on machine $A$ and 3 hours on machine $B$. The second type requires 3 hours on machine $A$ and 2 hours on machine $B$. Machines $A$ and $B$ can work at most 18 hours and 15 hours per day respectively. He earns a profit of ₹ 30 and ₹ 25 per trunk of first and second type respectively. How many trunks of each type must he make each day to make maximum profit? OR Two tailors $A$ and $B$, earn ₹ 300 and ₹ 400 per day respectively. $A$ can stitch 6 shirts and 4 pairs of trousers per day while $B$ can stitch 10 shirts and 4 pairs of trousers per day. How many days should each of them work if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost?

Translate the production limits into linear inequalities, plot the feasible region, evaluate the objective (profit) function $Z$ at every corner point, and pick the corner that maximises $Z$.

Step 1 — Formulate the LPP

Let $x=$ number of type-1 trunks made per day, $y=$ number of type-2 trunks made per day.

Machine A (3 h per type-1 trunk, 3 h per type-2 trunk, available 18 h):

$3x+3y\le18\ \implies\ x+y\le6$

Machine B (3 h per type-1 trunk, 2 h per type-2 trunk, available 15 h):

$3x+2y\le15$

Non-negativity: $x\ge0,\ y\ge0$.

Objective: maximise profit

$Z=30x+25y$

Step 2 — Find the corner points of the feasible region

Intercepts of $x+y=6$: $(6,0)$ and $(0,6)$.

Intercepts of $3x+2y=15$: $(5,0)$ and $(0,7.5)$.

Solve the two lines simultaneously to get their intersection:

$x+y=6\ \implies\ y=6-x$

$3x+2(6-x)=15\ \implies\ 3x+12-2x=15\ \implies\ x=3,\ y=3$

Testing which intercepts lie inside the other constraint: $(5,0)$ satisfies $x+y=5\le6$ ✓, and $(0,6)$ satisfies $3(0)+2(6)=12\le15$ ✓. So the feasible region is the quadrilateral with vertices

$(0,0),\ (5,0),\ (3,3),\ (0,6)$

Step 3 — Evaluate $Z$ at each vertex

The maximum value of $Z$ is $165$, attained at $(3,3)$.

Step 4 — Verify feasibility of $(3,3)$

$x+y=6\le6\ \checkmark\qquad 3(3)+2(3)=15\le15\ \checkmark$

Both machine constraints are exactly met (fully utilised), consistent with the optimum of a linear program occurring at a vertex where constraints are tight.

…