Linear programming: Sam and Jim have a small bakery that mak

guitarkid583

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Hey guys, this is a little long, but much help is appreciated.

Sam and Jim have a small bakery that makes two types of cookies: plain and iced. They need to decide how many dozens of each kind of cookies to make. One dozen plain cookies requires 1 pound of cookie dough and one dozen iced cookies requires .7 pounds of dough. Each dozen of iced cookies uses .4 pounds of icing. The plain cookies take .1 hours to prepare, and the iced cookies take .15 hours to prepare. They know all the cookies will be sold.

Limitations: They can only make up to 110 pounds of cookie dough. They have the ingredients to make up to 32 pounds of icing. They have a total of 15 hours for preparing the cookies.

X= number of dozens of plain cookies Y= number of dozens of iced cookies.

This is what I have done so for for the equations (tell me if I'm wrong):
Use of cookie dough: x + .7y ≤110
Use of icing: .4y ≤ 32
Preparation time: .1x + .15y ≤ 15
Number of cookies = x ≥ 0, y ≥ 0

Part 2

Jim and Sam sell the plain cookies for $6 a dozen and it costs $4.50 to make those cookies. The iced cookies sell for $7 a dozen and cost $5 a dozen to make. What is the profit expression for the cookies?

Part 3
Then I have to graph this, but I'm unsure of how to find the corner points or how to graph the equations themselves.

Thanks so much
 
Re: Linear programming: Sam and Jim have a small bakery that

guitarkid583 said:
The plain cookies take .1 hours to prepare, and the iced cookies take .15 hours to prepare.
They have a total of 15 hours for preparing the cookies.
What's wrong with that picture, Elvis?
If you can cook 'em all in .1 + .15 = .25 hour (or 15 minutes) , why the 15 hours? :shock:
 
Re: Linear programming: Sam and Jim have a small bakery that

Hello, guitarkid583!

Sam and Jim have a small bakery that makes two types of cookies: Plain and Oced.
They need to decide how many dozens of each kind of cookies to make.

One dozen Plain requires 1 pound of dough
. . and one dozen Iced requires 0.7 pounds of dough.
Each dozen of Iced uses 0.4 pounds of icing.
The Plain take 0.1 hours (per dozen) to prepare,
. . the Iced take 0.15 hours (per dozen) to prepare.

Limitations
They can only make up to 110 pounds of dough.
They have the ingredients for 32 pounds of icing.
They have a total of 15 hours for preparing the cookies.

x = number of dozens of plain cookies
y = number of dozens of iced cookies.

This is what I have done so far . . .

Use of cookie dough: \(\displaystyle \,x\,+\,0.7y\:\leq\:110\)
Use of icing: \(\displaystyle \,0.4y\:\leq \:32\)
Preparation time: \(\displaystyle \,0.1x\,+\,0.15y\:\leq\:15\)
Number of cookies: \(\displaystyle \,x\:\geq\:0,\;y\:\geq\:0\)

Looks good to me!



Part 2
Jim and Sam sell the Plain cookies for $6 a dozen and it costs $4.50 to make those cookies.
The Iced cookies sell for $7 a dozen and cost $5 a dozen to make.
What is the profit expression for the cookies?

The profit for Plain cookies is: \(\displaystyle \,\$6.00\.-\.4.50\:=\:\$1.50\) per dozen.
The profit for Iced cookies is: \(\displaystyle \,\$7.00\.-\.5.00\:=\:\$2.00\) per dozen.

The profit function is: \(\displaystyle \:p\;=\;1.5x\,+\,2y\)



Part 3
Graph the restrictions.

We have: \(\displaystyle \,x\,\geq\,0,\;y\,\geq\,0\)
. . Hence, we are in the first quadrant.

We have: \(\displaystyle \,0.4y\,\leq\,32\;\;\Rightarrow\;\;y \,\leq\,80\)
. . We have the horizontal line: \(\displaystyle y\,=\,80\)
. . Shade the region below it.

We have: \(\displaystyle \,x\,+\,0.7y\,\leq\,110\;\;\Rightarrow\;\;10x\,+\,7y\:\leq\:1100\)
. . The line \(\displaystyle 10x\,+\,7y\:=\:1100\) has intercepts \(\displaystyle (110,\,0)\) and \(\displaystyle \left(0,\,\frac{1000}{7}\right)\)
. . Graph the line and shade the region below it.

We have: \(\displaystyle \,0.1x\,+\,0.15y\:\leq\:15\;\;\Rightarrow\;\;2x\,+\,3y\:\leq\:300\)
. . The line \(\displaystyle 2x\,+\,3y\:=\:300\) has intercepts \(\displaystyle (150,\,0)\) and \(\displaystyle (0,\,100)\)
. . Graph the line and shade the region below it.

The graph looks like this:
Code:
      |
1000/7*
      | *
      |   *
      |     *
  100 o       *
      |   o     *
   80 @ - - - @ - * - - - - - - - 
      |:::::::::::o * 
      |:::::::::::::::@
      |:::::::::::::::::* o
      |:::::::::::::::::::*   o
      |:::::::::::::::::::::*     o
      |:::::::::::::::::::::::*       o
      @ - - - - - - - - - - - - @ - - - - o - -
      |                        110       150

We are concerned with the vertices of this polygon, indicated with @.

For some we must find the intersection of two lines.
. . You can do that, right?

Reading clockwise from the origin, the vertices are:
. . \(\displaystyle (0,\,0),\;(0,\,80),\;(30,\,80),\;(75,\,50),\;(110,\,0)\)

 
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