Linear Programming HELP!

[tonsoffun138]

I have not idea how to solve this....

Supose you make and sell skin lotion. A quart of regular lotion contain 2 c oil and 1 c cocoa butter. A quart of extra rich skin- lotion contain 1 c of oil and 2 c cocoa butter. You will make a profit $10/qt on reg. lotion and a profit $8/qt on extra rich lotion. You have 24 c oil and 18 c cocoa butter.

a. How many quarts of each type of lotion should you make to maximize ytour profit?

b. What is the maximum profit?

 

[soroban]

Hello, tonsoffun138!

I have no idea how to solve this.... You've never done one of these before?

Supose you make and sell skin lotion.
A quart of regular lotion contain 2 c oil and 1 c cocoa butter.
A quart of extra rich skin- lotion contain 1 c of oil and 2 c cocoa butter.
You will make a profit $10/qt on reg. lotion and a profit $8/qt on extra rich lotion.
You have 24 c oil and 18 c cocoa butter.

a. How many quarts of each type of lotion should you make to maximize your profit?

b. What is the maximum profit?

Let x = number of quarts of Regular lotion. .$\displaystyle x\:\geq\:0$ .[1]
Let y = number of quarts of Extra Rich lotion. .$\displaystyle y \:\geq \;0$ .[2]

Each quart of Reguar takes 2 unit of oil. .$\displaystyle 2x$ units of oil are needed.
Each quart of Extra takes 1 unit of oil. .$\displaystyle y$ units of oil are needed.
The total oil used is: $\displaystyle 2x\,+\,y$ units.
But we have only 24 units of oil: . $\displaystyle 2x + y \:\leq \:24$ .[3]

Each quart of Regular takes 1 unit of cocoa butter. .$\displaystyle x$ units of cocoa butter are needed.
Each quart of Extra takes 2 units of cocoa butter. .$\displaystyle 2y$ units of cocoa butter are needed.
The total cocoa butter needed is: $\displaystyle x\,+\,2y$ units.
But we have only 18 units of cocoa butter: .$\displaystyle x\,+\,2y \:\leq \:18$ .[4]


We will graph the four inequalities.

The first two, [1] and [2], place us in quadrant 1.

[3] $\displaystyle 2x\,+\,y\:\leq\:24$. .Graph the <u>line</u>: $\displaystyle 2x\,+\,y\:=\:24$
. . It has intercepts: (12,0), (0,24). .Shade the region below the line.

[4] $\displaystyle x\,+\,2y\:\leq\:18$. . Graph the line: $\displaystyle x\,+\,2y\:=\:18$
. . It has intercepts: (18,0), (0,9). .Shade the region below the line.

The region is a quadrilateral in the first quadrant.
. . We are concerned with the <u>vertices</u> of this region.
. . Three of them are obvious: (0,0), (12,0), (0,9).
. . To find the fourth, solve: $\displaystyle 2x\,+\,y\:=\:24$ and $\displaystyle x\,+\,2y\:=\:18$
. . . . and we get: $\displaystyle x = 10,\;y = 4$

We have four vertices to test: .$\displaystyle (0,0),\,(12,0),\,(0,9)\,(10,4)$

Test them in the profit function: $\displaystyle P \:= \:10x + 8y$
. . and see which pair produces maximum profit.