Linear Programming Word Problem

Question

A candy company has 145 lbs of cashews and 195 pounds of peanuts which can be combined into 2 mixed. The deluxe mix is 1/2 cashew and 1/2 peanuts which sells for $6 per pound. The economy mix is 1/3 cashew and 1/2 peanuts and sells for $4.30 per pound. How many pounds of each mix should be prepared to maximize revenue?

Answer 1

X is the amount of the deluXe mix and Y is the amount of the economY mix

 

1/2X + 1/3Y = 145

 

1/2X + 1/2Y = 195

 

 

The revenue function is F(x,y) = 6X + 4.30Y

 

 

1/2X + 1/3Y = 145

3x + 2Y = 870 <--- multiplies everything by 6

 2Y = 870 - 3x

 

 Y = (870 - 3x)/2 = 435 - 3/2X = -3/2X + 435

1/2X + 1/2Y = 195

X + Y = 390 <--- multiplies everything by 2

X = 390 - Y

Also Y = 390 - x = -x + 390

 

 

The lines meet at :

3(390 - Y) + 2Y = 870

 

1170 - 3Y + 2Y = 870

 

1170 - Y = 870

 

Y = 300 --->X = 90

So the lines meet at (90,300)

 

The graph of the critical region is bounded above
by the line y = -x + 390, to the right by y = -3/2X+435,

below by the x-axis, and to the right by the y-axis.

As usual, all quantities are positive, so the critical

region is in the first quadrant.

 

The critical points are (0,390), (90,300), and

(290,0) which is the x-intercept of where the

line y=-3/2X+ 435 crosses the x axis (the zero)

 

Plugging them into the revenue funciton 

(0,390) ---> 0x6 + 390*4.30 = 1677

 

(90,300) ---> 90x6 + 300x4.30 = 180 + 1290 = 1830

 

(290,0) ---> 290*6 = 1740

 

 

90 pounds of the deluxe mix, and 300 pounds

of the economy mix

The graph of the critical region is bounded above

by the line y = -x + 390