How Can I Correctly Set Up Equations for Graphing Production Constraints?

[fellixombc]

So I need help with setting up the equation to graph it (eg. y = mx + b) but with different variables.


In a factory, Machine A Produces 60 Cornflakes boxes per hour and Luckycharms at 70 boxes per hour. Machine B produces produces 40 cornflake boxes per hour and Luckycharms 40 per hour. It costs $50 a hour to run machine A and $30 a hour to run Machine B. Machine A cannot run more then 9 hours a day and Machine B cannot run more then 10 hours a day. Atleast 240 cornflakes boxes need to be made and 140 lucky charm boxes need to be made.

So, I setup my variables:

Cornflakes = C
Luckycharms = L


And here are my equations (there cannot be more than 4 constraints):

60C + 70L <= 9 (machine A)
40C + 20L <= 10 (machine b)
C >= 240
L >= 140


My equation is wrong though. If I solve and graph it (L = Y, C = X), it does not come out correctly.

Can someone help?

 

[jbunniii]

Your first two inequalities are wrong.

If C and L are the number of boxes of each type produced, then what is 60C? What is 70L? What are the units of these things? 60 = number of boxes per hour, C = number of boxes, so the units of 60C are (number of boxes)^2 / hour. This is not what you want.

Also, if C and L represent the TOTAL number of boxes of each type produced, then you seem to be assigning this total number to be produced by both machine A and machine B. Thus you will operate the machines longer than needed.

What if instead of C and L, you were to use two variables A and B, representing the number of hours of operation of machines A and B?

Then you would have these constraints:

\begin{align*} A &amp;\leq 9 \\<br /> B &amp;\leq 10 \\<br /> 60A + 40B &amp;\geq 240 \\<br /> 70A + 40B &amp;\geq 140\end{align*}

This assumes that all the production has to take place in one day. Does it?

Also, what additional constraint do the operating costs impose? Is the goal to choose A and B to minimize the total cost? (I assume so.) Then you will have to express this cost in terms of A and B and work out how to minimize it.

 

[fellixombc]

yes, i need to find the most miminzied cost, so i would graph this.

 

[jbunniii]

Sorry, my initial response assumed that each machine could produce box C *or* box L at a certain rate per hour, but on closer reading I believe you mean that each machine can produce *both* types simultaneously, at the stated rates. I've modified my original post accordingly. (You might need to hit "refresh" to see it.) Is that interpretation correct?

 

[fellixombc]

Sorry, my initial response assumed that each machine could produce box C *or* box L at a certain rate per hour, but on closer reading I believe you mean that each machine can produce *both* types simultaneously, at the stated rates. I've modified my original post accordingly. (You might need to hit "refresh" to see it.) Is that interpretation correct?

Oh i see now, yes it is. Thank you very much, I have a huge headache considering I've been working on this for 2 hours =/

Time for sleep and I will finish the rest of my assignment tomorrow. Thank you.

 

[jbunniii]

Oh i see now, yes it is. Thank you very much, I have a huge headache considering I've been working on this for 2 hours =/

Time for sleep and I will finish the rest of my assignment tomorrow. Thank you.

No worries, and good luck.

By the way, if you really want to use C and L instead of A and B, you can certainly do so, by solving these equations for A and B:

\begin{align*}60A + 40B &amp;= C \\<br /> 70A + 40B &amp;= L \end{align*}

Then the last two inequalities become

\begin{align*} C &amp;\geq 240 \\<br /> L &amp;\geq 140 \end{align*}

and the first two inequalities end up looking pretty weird and unintuitive. I think it's more natural to use A and B as I defined them.