Let x units of product A and y units of product B be produced and sold by the carpenter. Then information given in the statement is tabulated as:
Then the LPP is maximise P = 48x + 40y
Subject to the constraints:
2x + y ≤ 90 …..(i)
x + 2y ≤ 80 ….. (ii)
x + y ≤ 50 ….. (iii)
x ≥ 0, y ≥ 0
Draw the graphs of equations (i), (ii), (iii)
Then shaded region is the required feasible region which is bounded with comer points
O(0, 0), A(45, 0), B(40, 0), C(20, 30) and D(0, 40).
At O(0, 0), the value of P = 0 + 0 = 0
At A(45, 0), the value of P = 48 × 45 + 0 = 2160
At B(40,10), the value of P = 48 × 40 + 40 × 10 = 2320 → Maximum
At C(20, 30), the value of P = 48 × 20 + 40 × 30 = 2160
At D(0, 40), the value of P = 0 + 40 × 40 = 1600
We have the maximum value of P as ₹ 2320 and it is obtained at the vertex B (40, 10).
Hence, the maximum gross income of the carpenter should make 40 units of product A and 10 units of product B.