00:01
You are the manager of bornees, a toy manufacturing company.
00:05
The cost function for your remote control cars is c of x.
00:09
And c of x is 0 .25 times x square plus 96x plus $800 per day.
00:21
And x here is the number of remote control cars you manufacture per day, up to a maximum of 3 ,000 cars per day.
00:33
Okay.
00:35
How many remote control cars should you build to minimize the average cost? so once we here minimize, maximize what we should be thinking is optimizations, all right? so this is an optimization question.
00:52
And for optimization, i developed a little bit of a recipe or you could call it an algorithm.
00:58
Very simple.
01:03
It might not be the most you know, comprehensive, but it should give us the most important things that we need to know in solving optimization functions.
01:19
Well, the first thing is to identify or develop the relationships.
01:24
And when i say relationships, i really mean functions, right? relationships is more general term to say what we're really trying to look for.
01:34
On the relationships, we could identify the objective function.
01:37
Now, the objective function is the function or if we have multiple objective functions, those are the functions that we're trying to optimize.
01:49
We're trying to either minimize or maximize them.
01:53
And this is the most important part, identifying the function we're trying to optimize.
01:58
The next thing would be the supporting functions.
02:01
Well, we could put this first, identify the supporting functions or the functions generally.
02:08
But it turns out sometimes we're supporting functions.
02:09
We could use the supporting functions to help us find objective functions or to find the constraints.
02:18
Or they would just give us some way to quantify certain parameters or quantities that we have.
02:27
So just identify the relationships.
02:30
So on this, the third one would be the constraint.
02:34
Sometimes they're explicitly given or sometimes they're implicitly given.
02:38
In this case, it was implicitly given in the question.
02:41
I interpreted it as follows.
02:44
So the line where we get this constraint from is where x is the number of remote control cars you manufacture per day up to a maximum.
02:53
So when you hear maximum there is x cannot be greater than 3 ,000.
03:01
So up to a maximum of 3 ,000 cars, meaning x is equal to or less than 3 ,000.
03:09
But on this other side, we can't, in this case, so you have to apply some, some, you know, common knowledge or just some sense into this.
03:24
Whatever makes sense, you cannot manufacture negative number of cars.
03:32
It doesn't make any sense.
03:33
It's either you're not manufacturing anything or you're manufacturing something.
03:38
And in this case, we have a maximum of 3 ,000.
03:41
So therefore, the minimum would be 0, right? so you can see that the constraint here would be x would be between 0 and 3 ,000.
03:49
Thousand.
03:54
Sometimes the constraint would be a little more implicit, you know.
04:01
So you have to watch out for that and you could have multiple constraint functions sometimes.
04:08
So we're done with the relationship.
04:09
So let's go ahead and, you know, deal with that.
04:13
First of all, we've already found the constraint here.
04:18
And the question is asking us to do what? minimize the average costs.
04:24
Now, this here is just costs.
04:28
It's a cost function.
04:30
It is not the average cost.
04:32
The first thing i want you to pick out is not the average cost.
04:35
It's just a cost function.
04:38
But we can find the average cost.
04:40
An average cost, which i would call c bar of x, the bar there should represent average.
04:49
So c bar of x would be equal to the cost function divided by x, the number of cars.
05:01
So if this represents, the total cost to manufacture x amount of cars, then the average cost, meaning the cost per car, would have to be the total number divided by the number of cars.
05:17
It's kind of like saying sum of items over number of items.
05:25
If you remember, this is one way to describe what average is, right? so the average cost function, c bar of x would be the cost.
05:39
Function c of x divided by x now if you notice this c of x therefore would be a supporting function here while c bar of x is the actual objective function that's what we want to minimize so we want to optimize right so let's find what c bar of x is c bar of x would just be 0 .25x square plus 96x plus 800 all of that divided by x what we could do here is you know break this apart into the partial fractions and what we would get would be x x divides 0 .25 x square and 96x divided by x plus 800 divided by x which would finally give us that c bar of x is 0 .25 times x square so thanks x not x square now plus 96 plus 800 divided by x so this is c bar of x this is our objective function this is what we're going to minimize this is the function we need to this is an objective function of optimization now one thing i want you to quickly see here is also that x now has to be in this range meaning x cannot be zero otherwise this function would not exist so x has to be under here...