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.
