00:01
Hi there.
00:02
In this problem, we have two different plants creating a product and q1 is the quantity produced by the first plant, q2 by the second.
00:11
So we're going to try to figure out the best values of q1 and q2 to maximize profit.
00:18
So we need to start by writing a formula for profit.
00:21
So profit will be this capital p here.
00:24
Let's start by remembering that profit equals revenue minus costs.
00:28
And furthermore, revenue equals this is a little p equals price times quantity that you're selling.
00:38
So if we just carefully substitute, we will be okay here.
00:45
So we will get, uh, no, instead of p, right, we have an expression for p.
00:52
So this p is 60 minus 0 .04, uh, q.
00:59
Now instead of q, remember q really equals q 1 .9 .1.
01:05
Plus q2.
01:06
That's the total quantity we're told.
01:08
So that's what i'll do here.
01:09
I'll make this q1 plus q2.
01:15
Okay.
01:16
That's that what we just did is just the p.
01:19
So we took care of this part.
01:20
Now times q and again, remember q is actually q1 plus q2.
01:27
So just so you can see what happened here, this first part is all little p price.
01:32
Second part is all q, the quantity that we're selling.
01:37
Okay.
01:40
Um, that, right, that's just our revenue.
01:43
We still have to subtract off the cost.
01:45
Cost functions, we have two of them, one for each plant.
01:48
So we'll have to subtract off both of them.
01:50
So we get minus 8 .5 minus 0 .03 q1 squared, minus 5 .2 minus 0 .04 q2 squared.
02:01
So all of this is subtracting off cost.
02:07
So that is our profit function.
02:09
We'll have to clean it up a little bit so that we can do something.
02:12
With it and and it's a lot of algebra among other things to be distributing q1 plus q2 to everything we'd foil everything out here combine a couple like terms here with the numbers so i'm going to put dot dot dot here because this is the calculus part that's the hard part here i'll assume that your algebra is okay but here you can check your answer against mine what you should end up with at least maybe in a different order, but what you should end up with at the end of the day is 60 q1 plus 60 q2 minus 0 .07 q1 squared minus 0 .08 q2 squared minus 0 .08 q2 squared minus 0 .08 q2 minus 13 .7.
03:12
Again, that just comes from algebra and arithmetic on this profit function.
03:17
Now it'll be easier to take derivatives though.
03:19
That's the only thing.
03:21
Okay, so that's our profit function.
03:23
The next thing we do, if you want to find a maximum, is find critical points.
03:29
So let's take partial derivatives.
03:31
Let's take p sub q1.
03:35
So the partial derivative of all this with respect to q1.
03:39
Okay, this first term will give us a derivative of 60.
03:42
This next term, zero, because there's no q1 in it.
03:46
The next one after that, we'll get minus.
03:50
0 .14 q1.
03:54
Next term, 0, because there's no q1.
03:57
Then finally, this second to last term, q2 is a constant, so we'll get minus 0 .08 q2...