caterer must make at least 50 gal of potato soup

Question

A caterer must make at least 50 gal of potato soup and at least 120 gal of tomato soup. One chef can make 5 gal of potato soup and 6 gal of tomato soup in 1h. Another chef can make 4 gal of potato soup and 12 gal of tomato soup in 1h. The first chef earns $20/h. The second chef earns $22/h.How many hours should the company ask each chef to work to minimize the cost? 

Answer 1

 

 5x + 4y = 50

 

 

 6x + 12y = 120 ---> x + 2y = 20 ---> x = 20 - 2y

 

 5(20 - 2y) + 4y = 50

 

 100 - 10y + 4y = 50

 

 100 - 6y = 50

 

 -6y = -50

 

 y = 50/6 = 25/3 

 

5x + 4(25/3) = 50

 

5x + 100/3 = 50

 

5x = 50 - 100/3 = 150/3  - 100/3 = 50/3

 

x = 10/3

 

Use the first chef for 3 hours and 20 minutes and

the second chef for 8 hours and 20 minutes

 

 

 

 Cost = 20X + 22Y

 

It will cost 20(3 and 1/3) + 22  ( 8 and 1/3) = 20( 10/3) + 22(25/3) = (200+ 550)/3 = 750/3 = 250