Answer: (b) 80
Solution:-
Let x and y be the number of packets of food P and Q respectively.
Obviously, x \(\geq\) 0 and y \(\geq\) 0. ( Because number of packets never be negative.)
Since, dietician wants to minimise the amount of cholesterol in the diet. Food P contains 2 units of cholesterol and Food Q contains 7 units of cholesterol.
Mathematical formation of the given problem is as follows:
The Objective function is minimise Z = 2x + 7y.
8x + 4y \(\leq\) 360 ( constraint on protein ), i.e. 2x + y \(\leq\) 90 ... (1)
6x + 12y \(\geq\) 240 ( constraint on vitamin A ), i.e. x + 2y \(\geq\) 40 ... (2)
2x + 2y \(\leq\) 144 ( constraint on calcium ), i.e. x + y \(\leq\) 72 ... (3)
x \(\geq\) 0 and y \(\geq\) 0 ( non negative constraint ) ... (4)
By changing inequalities (1) to (4) into equations, we get
2x + y = 90 ... (5)
x + 2y = 40 ... (6)
x + y = 72 ... (7)
x = 0 and y = 0 which is y-axis and x − axis respectively ... (8)
Now, the intersection point of equation 2x + y = 90 with x-axis and y-axis are A(45, 0) and F(0, 90), respectively.
Now, intersection point of equation x + 2y = 40 with x-axis and y-axis are E(40 , 0) and D(0, 20), respectively.
Now, intersection point of equation x + y = 72 with x-axis and y-axis are G(72 , 0) and C(0, 72), respectively.
Now, intersection point of equations 2x + y = 90 and x + 2y = 40 is H\(\big(\frac{140}{3},\frac{10}{3}\big).\)
Intersection point of equations 2x + y = 90 and x + y = 72 is B(18, 54).
And intersection point of equations x + 2y = 40 and x + y = 72 is (104 , −32).
Since, point O(0, 0) satisfies inequality 2x + y \(\leq\) 90. Therefore, the feasible region (shaded region) includes the point O(0, 0) with respect to the inequality (1).
Since, point O(0, 0) does not satisfies inequality x + 2y \(\geq\) 40. Therefore, the feasible region (shaded region) does not include the point O(0, 0) with respect to inequality (2).
Since, point O(0, 0) satisfies inequality x + y \(\leq\) 72. Therefore, the feasible region (shaded region) includes the point O(0, 0) with respect to inequality (3).
By inequalities x \(\geq\) 0, y \(\geq\) 0 given in equation (4) we mean first quadrant in graph.
The drawn graph with feasible region (shaded region) is given below
The shaded region ABCDE is the feasible region determined by the system of constraints given in equations (1), (2), (3) and (4). We noticed that feasible region ABCDE is bounded and the corner points of feasible region are A(45, 0), B(18, 54), C(0, 72),D(0, 20) and E(40, 0).
We know that if feasible region is bounded then the optimal value (maximum or minimum value) must occur at a corner point of the feasible region.
| Corner point |
Objective function
min Z = 2x + 7y |
|
A (45,0)
B (18,54)
C (0,72)
D (0,20)
E (40,0) |
90
414
504
140
80 |
From the table, we find that the minimum value of the objective function z is 80 at the point E(40, 0) of the feasible region.
Hence, the maximum value of the objective function z is 80.
Therefore, the minimum amount of cholesterol is 80 in the diet.
