00:01
All right, friends, let's take a look at this problem together.
00:03
So the situation that we have going on is a manufacturer.
00:07
And this particular manufacturer is manufacturing two types of products.
00:12
They are manufacturing a product a and a product b.
00:17
So let's go and write this.
00:19
We are talking about product a and product b.
00:29
And so there are two different processes.
00:32
That it takes to make these products.
00:35
Process 1 and process 2.
00:43
So process 1 takes 6 hours for part a, product a, and 12 hours for product b.
00:53
Process 2 takes 5 hours for product a and 4 hours for product b.
01:00
The machine that uses process one, the machine that uses product or process one, excuse me, has a total of 75 hours that are available to that process.
01:20
And then process two has a different machine and it can run for 55 hours.
01:28
And so this is per week.
01:33
And in this particular situation, what we understand is that the profit that you are able to make is going to be $12 for product a and $10 for product b.
01:51
Now, before we do anything else, i do want you to make some observations about what you see here.
01:58
So product a takes a total of 11 hours for creation.
02:05
So six hours here, five hours here, total of 11 hours.
02:08
Product b, total number of hours, 12, 13, 14, 15, 16 hours.
02:16
So also notice that you make more profit with process a than you do with process b.
02:25
So typically when you are making, well, let's just write out product a.
02:32
So it's more profit, more profit, more money, less time.
02:41
So it seems like a better deal to have process a happen more times possibly than process b.
02:51
So process b takes less time, or excuse me, even more time.
02:58
Whoops, let's write that out.
03:00
Less money is what i meant.
03:03
Less money and more.
03:05
Time.
03:07
So it seems like you would make more profit if you had more of type a.
03:13
However, a good combination to making sure that you're using both systems at your disposal at your manufacturer is important.
03:21
So what we are going to do is to graph these two equations.
03:25
And the two equations that i'm going to graph are these two.
03:29
And the way that i'm going to graph these is to convert them into an x and a y.
03:34
So i'm going to say six x.
03:37
Plus 12y is less than or equal to 75.
03:40
And then i'm going to say 5x plus 4y is less than or equal to 55.
03:47
So as we do that, let's take a look at our graphs that we have available to us.
03:54
So this is our graph of our very first equation.
03:58
Let's go ahead and rewrite those equations so that we can understand our work.
04:02
This is the equation of just 6x plus 12y is less than or equal to 75.
04:08
Now, if you wanted to see this in slope intercept form, you could modify the equation, but for right now i'm not going to worry about that.
04:16
This equation is adding, so we're adding the green line that is the 5x plus 4y is less than or equal to 55, and this one is in green.
04:29
And this, of course, is in red.
04:31
So we've graphed the two equations, and what we see when we graph these two equations is that there are some critical points that we have to look at.
04:40
And so i'm going to put some dots here where our critical points are.
04:46
Critical points are where our double shaded or multiple areas of shaded regions are.
04:54
So in this case we have four points that bound our double shaded region, and the first one is zero -zero...