Question

Linear Programming:
Carpentry: Emilio has a small carpentry shop where he makes large and small bookcases. His profit on a large bookcase is $80 and on a small book case it is $50. It takes Emilio 6 hours to make a large bookcase and 2 hours to make a small one. He can spend only 24 hours each week on his carpentry work. He must make at least two of each size each week. What is the maximum weekly profit?

Answer 1

Wait, so we don't know the profit of a small book case?

Answer 2

Hold up, i screwed the question, ima edit it now

Answer 3

There.

Answer 4

So let's write out some equations.
\[p=80l+50s\]\[6l+2s \leq 24\]\[l\geq 2\]\[ w\geq2\]

Answer 5

Why did you put 'Linear Programming' up there?

Answer 6

Because that is what this is. And i dont understand how you got the equations.

Answer 7

\(p\) is just our profit equation

Answer 8

Oops, \(w\geq 2\) should be \(s \geq 2\)

Answer 9

lol, i was like where did the w come out of.

Answer 10

Where as \(s\) and \(l\) are just the number of each bookcase.

Answer 11

Could we use x and y if you dont mind, i can understand better with those variables.

Answer 12

Is this a programming course?

Answer 13

Its precal

Answer 14

Do you know how to graph inequalities?

Answer 15

Yes. We were given a graph to plot points on too.

Answer 16

If you let \(l=x\) and \(s=y\) then you can graph and get all the possible solution sets which fit the inequalities... but that still won't give you the max.

Answer 17

Tough question for precalc

Answer 18

lol, so when i graph these, what do i do?

Answer 19

What I would do actually is.... let \(x = l-2\) and \(y=s-2\). Then we get:
\[6(x+2) + 2(y+2) \leq 24\]\[ x\geq 0\]\[y \geq 0\]

Answer 20

How did you do that?

Answer 21

Then we get: \[6x + 12 +2y + 4 \leq 24 \Rightarrow 6x+2y \leq 8\]

Answer 22

Well if I set \(x = l-2\) then \(l=x+2\) if we want to replace \(l\) with \(x\)

Answer 23

Wait, in still a little confused, how did you have x=1-2 and y=s-2?

Answer 24

I'm sorry for confusing you... Since \(x\) and \(y\) are variables we are bringing into this equation, we can set them equal to whatever we like.

Answer 25

The reason I chose \(l-2\) is because \(l \leq 2 \) gives us \(l-2\leq 0 \)

Answer 26

And \(\leq 0\) is easier to deal with than \(\leq 2 \)

Answer 27

Cant we just graph those?

Answer 28

Sure

Answer 29

Its much easier to graph than solve it all.

Answer 30

Depends on the person, but do whatever you think is easier.

Answer 31

\[x \le2\] is pretty easy to graph. So i would graph that alongside the other equations?

Answer 32

Basically, graph all of the inequalities.

Answer 33

\[6x+2y \le24\]
How did you get that?

Answer 34

Solve for \(y\).

Answer 35

No, i mean, how did you recieve that equation?

Answer 36

Because the large ones take 6 hours, the small ones take 2 hours and you can only spend 24 hours a week.

Answer 37

Ohhh okayy, got it. Thanks for the explanation. So once we graph this, it should make a shape, in which we will take the vertices, and plug them into the Profit equation?

Answer 38

You will get a shape, you want to take every coordinate in the shape (that is a whole number) and plug them into the profit equation.

Answer 39

And once i get that coordinate, i will have to plug each one in to find the largest amount which will be the maximum weekly profit right?

Answer 40

Yes

Answer 41

Thanks soo much.