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in Linear Programming by (58.3k points)

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Types of Toys Machines
I II III
A 12 18 6
B 6 0 9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

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Best answer
         Machines
Toys I II III Profit
A(x) 12 18 6 Rs. 7.50
B(y) 6 0 9 5

Maximize

z = \(\frac{15}{2}\)x + 5y

subject to the constraints 

(i) 12x + 6y ≤ 360 

⇒ 2x+y ≤ 60 

(ii) 18x ≤ 360 ⇒ x ≤ 20 

(iii) 6x + 9y ≤ 360 

⇒ 2x +3y ≤ 120 

(iv) x ≥ 0 , y ≥ 0

ABCDE is the solution region 

A (0, 0) Z = 0 

B (20, 0) Z = Rs 150 

C (20, 20) Z = Rs 250 

D (15, 30) Z = Rs 262.5 , 

E (0, 40) Z = Rs 200 

At D (15, 30) Z is maximised to Rs 262. 5

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