1.

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/ cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lam. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hour and the grinding/ cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs. 5 and that from a shade isRs. 3. Assuming that the manufacturer can sell all the lamps and shadesthat the produces how should be scehdule his daily production in order to maximise his profit?

Answer»

Solution :Let the MANUFACTURE makes `x` pedestal lamps and `y` wooden shades per day then

Maximise `Z=5x+3y`…………. 1
Constraints `2x+yle12`…………………2
`3x+2yle20`………………..3
`xge0,yge0`………………….4
First, draw the graph of the equation `2x+y=12`

Put `(0,0)` in the inequation `2x+yle12`,
`2xx0+0le12`
`0le12` (True)
THEREFORE, half plane contain the origin.
Since `x,yge0`
Therefore FEASIBLE region will be in first quadrant.
Now, draw the graph of the line `3x+2y=20`

Put `(0,0)` in the inequation `3x+2yle20`
`3xx0+2xx0le20implies0le20` (True)
Therefore half plane contains the origin.

From equations `2x+y=12` and `3x+2y=20`
The point of intersection is `B(4,4)`
Therefore the feasible region is OABCO.
Its vertices are `O(0,0),A(6,0),B(4,4)` and `C(0,10)` we FIND the value of `Z` at these vertices.

The maximum value of `Z` is Rs. 32 at point `B(4,4),`. Therefore 4 pedestal LAMP and 4 wooden shade should be made the manufacturer to obtain the maximum profit.


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