I need to know the variables (x,y), inequality statements and the optimization statement for the following,?Rainbow Paints had 1000 gallons of...

Resources:Yellow paint 1000gal.Red paint 1250 gal

Mixture1: Ratio of yellow:blue = (1/2):((1/2)

Mixture2:Ratio of yellow :Blue= (1/3):(2/3)

Let the Mixture1 and Mixture2  be x  gal  and y  gallons.

The  yellow paint in the two mixtures = x/2+y/3  gallons canot exceed  the company's stock of 1000gal.Similarly the blue pain quantity in two mixtures x/2+2y/3 gal cannot exceed 1250 of stock. I.e.'

x/2+y/3<=1000                                     (1)

x/2+2y/3<=1250                                   (2)

By capcity of production, x+y =<2150       (3)

By demand first mix x>=250                     (4)

and second mix y>=150.                         (5)

The profit  P  for the quantity of x and y at the  rate of $7.82 per unit of x and $6.67 per unit of y is given by:

P= 7.82x+6.67y                                   (6)

The object is to maximise  the profit P and find the quantities x and y satisfying the conditions or constraints (1) to (5).

Solution:

From (1) and (2):  Treating like ordinary equations  we find  x and y

(2)-(1): y/3=1250-1000==> y=750. So, y<=750 is a possibility.

From this  treating like simultaneous equations, we get by substitution in(1): x/2=100-250=750=>x=1500

So, x<=1500 is a possible solution

From these two possible solutions we get

X+y<=1500+750=2250 or

x+y<=2250 (7)

From (3) x+y < =2150. (8).

The conditions (7) and (8) should hold good and the satisfying solution should maximise profit,P.Therefore, we decide to keep x at 1500 without reducing the quantity as the rate of profit per unitof x is $7.82 which is higher than that of y at $6.66 and limiting  the y units to 650 as x+y=2150 at this value satisfying all the consraints, giving best or maximum profit.

P=7.82*1500+ 650*6.67 = $16,065.50, he maximum profit.

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