Answer:
Let the number of fans purchased be x and the number of radios purchased by y. \[\therefore \] The objective function id maximize \[Z=22x+18y\] Subject to the constraints \[360x+240\,y\le 5760\] or \[3x+2y\le 48,\] [on dividing both sides by 120] \[x+y\le 20\] and \[x,\,\,y\ge 0\] On considering the constraints as equation, we get \[3x+2y=48\] ? (i) \[x+y=20\] ? (ii) x = 0, y = 0 ? (iii) Tablet for the line \[3x+2y=48\] or \[y=\frac{48-3x}{2}\] is
So, line \[3x+2y=48\] passes through the points (0, 24) and (16, 0). On putting (0, 0) in the inequality \[3x+2y\le 48,\] we get \[3(0)+2(0)\le 48\Rightarrow 0\le 48,\] which is true. So, the half plane is towards the origin. Table for the line \[x+y=20\] or \[y=20-x\] is x 0 16 y 24 0
So, line \[x+y=20\] passes through the points (0, 20) and (20, 0). On putting (0, 0) in the inequality \[x+y\le 20,\]we get \[0+0\le 20\] \[\Rightarrow \] \[0\le 20,\] Which is true. So, the half plane is towards the origin. The graphical representation of these lines is given below The point of intersection of lines (i) and (ii) is B (8, 12). The shaded region id the graph represents the feasible region and its corner points are O (0, 0), A (16, 0), B (8, 12) and C (0, 20). Now, the values of Z at corner points are given below x 0 20 y 20 0
So, Z is maximum at x = 8 and y = 12 and the maximum value of is Rs. 392. Hence, the dealer should invest \[8\times 360=\,\,Rs.\,2880\] and \[12\times 240=\,\,Rs.\,2880\] in fan and radio respectively, to have a maximum profit of Rs. 392. Corner points \[\mathbf{Z = 22x + 18y}\] O(0, 0) \[Z=22(0)+18(0)=0\] A(16, 0) \[Z=22(16)+18(0)=352\] B(8, 12) \[Z=22(8)+18(12)=392\](maximum) C(0, 20) \[Z=22(0)+18(20)=360\]
You need to login to perform this action.
You will be redirected in
3 sec