A furniture trader deals in tables and chairs. He has Rs. 75,000 to invest and a space to store at most 60 items. A table costs him Rs. 1,500 and a chair costs him Rs. 1,000. The trader earns a profit of Rs. 400 and Rs. 250 on a table and chair, respectively. Assuming that he can sell all the items that he can buy, then which of the following is/are true for the above problem.
(A) Let the trader buys \(x\) tables and \(y\) chairs. Let Z denote the total profit. Thus, the mathematical formulation of the given problem is
Maximize \(Z=400x+250y\)
subject to constraints: \(x+y \le 60\), \(3x+2y \le 150\), \(x \ge 0\), \(y \ge 0\)
(B) The corner points of the feasible region are (0, 0), (50, 0), (30, 30) and (0, 60).
(C) Maximum profit is Rs. 19,500 when trader purchase 60 chairs only
(D) Maximum profit is Rs. 20,000 when trader purchase 50 tables only
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B), and (D) only
With x tables and y chairs, the constraints are:
\(\text{space: } x + y \le 60\)
\(\text{money: } 1500x + 1000y \le 75000, \text{ that is } 3x + 2y \le 150\)
\(x \ge 0,\ y \ge 0\)
and the profit to be maximised is Z = 400x + 250y.
The two lines x + y = 60 and 3x + 2y = 150 meet where
\(3x + 2(60 - x) = 150\)
\(x = 30,\ y = 30\)
On the x-axis the money line gives (50, 0) (the space line’s (60, 0) costs too much); on the y-axis the space line gives (0, 60) (the money line’s (0, 75) needs too much space). So the corner points are (0, 0), (50, 0), (30, 30), (0, 60), with
\(Z(0, 0) = 0\)
\(Z(50, 0) = 20000\)
\(Z(30, 30) = 12000 + 7500 = 19500\)
\(Z(0, 60) = 15000\)
So the maximum profit is Rs. 20,000, at (50, 0), that is 50 tables and no chairs.
Checking the statements
(A) Let the trader buys x tables and y chairs. Let Z denote the total profit. Thus, the mathematical formulation of the given problem is Maximize Z = 400x + 250y subject to constraints: x + y ≤ 60, 3x + 2y ≤ 150, x ≥ 0, y ≥ 0 — Correct. The money constraint 1500x + 1000y ≤ 75000 divided by 500 is 3x + 2y ≤ 150.
(B) The corner points of the feasible region are (0, 0), (50, 0), (30, 30) and (0, 60). — Correct. The lines meet at (30, 30), and the axis points are (50, 0) and (0, 60).
(C) Maximum profit is Rs. 19,500 when trader purchase 60 chairs only — Incorrect. 60 chairs give Z(0, 60) = 15,000, and the maximum is 20,000.
(D) Maximum profit is Rs. 20,000 when trader purchase 50 tables only — Correct. Z(50, 0) = 20,000 is the largest corner value.