Hello!
Mr Tarrou again, thank you for watching my videos, I do really appreciate it, and I hope
your finding them to be helpful.
Well today we're going to talk about one of my favorite subjects in math, but it can also
be one of the most frustrating, and that is word problems; modelling with functions.
You know, this is really where you start to see some of this math get put into real life,
and maybe not just anyone doing these types of problems, but someone; whether you work
at a corporation doing big projects or your an engineer, or just a builder, these word
problems, actually do have an application and purpose for some of day to day life.
also with word problems, we don't talk in terms of x's and y's, we don't use that language
in real life.
So, if you write an equation in plain language, you might notice that you do a lot more algebra
than you really think, in your everyday life.
But let's be honest a lot of it is just addition, subtraction and a little bit of percents when
you deal with money.
These can be frustrating because we don't deal with them often and word problems are
always hard to set up.
But they are getting applied, all this algebra is getting applied to some degree of reality,
which can make things seem a bit complicated.
So if you have a business, you'd like to be able to track and predict your revenues and
your sales based on certain parameters that are already happening in your business, and
let's take a look at that first with this first word problem.
Um, I'm just sort of giving you the facts of the problem instead of writing a big long
thing out and covering this whole board.
Let's say that your selling something right now and your selling about 4,500 units at:
$50/piece.
Maybe these are concert tickets, maybe just something your selling at an electronics store…
whatever, but your selling 4,500 units of something whether that's per day, per week,
per month or something, but it's a set value or set amount and it's $50 a piece.
Now whether a statistician has helped you out or maybe this is a survey from your customers...
But we are thinking that if I raise this price by... if I raise this price a dollar.
So we are going to raise this price by a dollar.
If I do that, we expect to lose... not loose but lose 5 sales... lose 5 sales or 5 units.
So we are going to use this information.
We are selling 4500 units of something for 50 dollars a piece.
If we raise that price by a dollar, for every dollar that we raise the price we are expecting
to lose 5 units of sales.
Can we come up with a formula that would model that situation.
Then take another step further and set up a formula that would calculate our overall
revenue.
Well let's just do one for the sales for the moment.
The number that you are selling based on the price that it is going to be equals...
Now all of your formulas for like say carbon 14 dating from science, carbon 14 dating where
they say you have something you found and it has a certain amount of carbon and then
so many years you lose half of it... or money in the bank.
You start with a certain amount and then hopefully every month that amount goes up.
Of course now days it is going up really really slowly.
Too slow you might think.
But formulas will generally start, that get applied to real life where you have this initial
value.
Well our initial value is our 4500 units of sales.
And we are expecting our sales to go down 5 every time we raise the price by a dollar
based on some kind of market research.
So we are going to lose 5 units of sales for every dollar that we mark up the price.
How do we bring that into this formula?
Well, let's say we are selling it for 50...
Let's say we sold it for 51?
Well wouldn't you just subtract 5 one time.
And what if the item were 52 dollars?
Well 52 dollars is two dollars over 50, so you would subtract 5 twice.
Well what if it was 55 dollars for the new price.
Wouldn't you subtract 5 dollars five times because your 5 dollars over?
So I am not really sure what goes in this formula, but I am making up some examples
which you should be able to do with word problems..or should do.. to help you see the pattern of
what is going on.
51, we subtracted this once.
52 dollars, we subtract this twice.
55 dollars, we are going to subtract this 5 times.
Now how is my brain coming up with those values?
Or how might you be coming up with those values?
Well I am subtracting.
I am subtracting the new price, which I am not using the variable of x and y to try to
make this make more sense, I am taking the new price and I am subtracting with the old
price.
So I can find how much is the increase.
55 dollars, the increase would be 5 dollars.
Ok, now you subtract by 5 five times.
This is it.
Our number items sold is going to be based on the original 4500 minus 5 times the difference
of the new price and the old price.
Again, we are going to lose... we think..
we are going to lose 5 sales for every one dollar of increase, not every one dollar in
the price.
Now if I clean this up a little bit I am going to have 4500 minus 5 times the price....
-5 times -50 is going to be... negative times negative is positive....
the parenthesis should not be there any more...
250.
Good, then 4500 plus 250 is 4750.
Good.
Now I am using full words, but I know you are not going to see that in your textbook.
So let's on the side say that x is going to be the new price of the item.
Well now this becomes -5 times x.
And this might look like in a textbook like n(x).
See, you loose a lot of the context of what is going on when you just start using x's
and y's or function notation as opposed to actual real plain language like we actually
use every day.
So this is the number of items being sold based on our assumption that the initial sales
of 4500 and that for every dollar of increase in price we will loose 5 units of sales.
Ok.
So that is the number of items being sold.
Now another common question in textbooks is, "What is your total revenue?"
Let me make you this simple example.
If I sell three items at $20 a piece wouldn't that be $60 dollars?
That would be a revenue of $60.
So revenue is simply how many you are selling times the price.
Well, how many of these items are we selling?
It is not 4500.
It is 4500 minus five times the increase in price.
Or, we just figured that out.
This is how many items we are going to sell based on past sales and our new price.
So the number is going to be 4750-5x times what?
Well if I kept in plain language I could have said the number sold times the price, but...
you know.... I know you are not going to see full words in your math book.
So this is the number of new items being sold based on the new price and the x is the price.
It is defined as the price of the items.
So it is the number sold times the price.
BAM!
There you go, revenue.
So if I distribute this x we get 4750x-5x^2.
Now I am going to go off subject a little bit or expand on this a bit more.
I will do these examples in two videos because I know I am not going to fit in all the ones
I need in just one.
If you were an economist or a businessman you would care about this a lot.
You would want to be able to find out the maximum or the best price to sell your item
and make the most money.
No one cares about how many seats are in the football stands or really how many items you
sell, it is all about the profit...
the revenue coming it.
Revenue and profit is not the same, but you want to maximize your revenue to maximize
ultimately your profit.
Now we are not doing this in my class, but if you wanted to maximize your revenue based
on the cost the item you have a perfect example and perfect situation where you can do this.
Anything with an exponent of two, just one exponent of 2, is a parabola.
And this parabola has a leading coefficient which is negative.
So if you were to graph this with your graphing calculator, and this will be covered in my
class somewhere else or maybe in your class, and I don't know exactly what this is going
to look like.. but I do know that this is a downward parabola because of the leading
coefficient.
The degree on the secondary term...
Did I just say degree?
The leading coefficient of the second degree term is negative.
So I know that this is going to be a downward opening parabola.
So if I were to graph it on a calculator, I would get a parabola like this somewhere.
And I could use my parabola to find my relative maximum.
When I find that point, well the x is the price $, the y is the revenue.
Now this is the price per item, and the y axis is going to revenue.
I can determine what would be the best price to sell my item to maximize my revenue.
Maybe it is not 50 dollars, maybe it is $55 or $60.
You might even find out that you need to lower the price, but that would be outside the scope
of this question because lowering the price would give a whole new different formula.
But there is an application for this if you are in economics where you want to find the
perfect price to maximize your revenue.
You can see that parabola happening here with the power of two.
So that is a little bit of expansion that you might do in a problem in other math classes.
My students right now are not, we are just setting up the formulas.
Ok.
Let me see if I can get another example done before I get out of here.
Let's say that I want to build a box.
And the box I am going to build is going to be an open box.
I am going to use a piece of cardboard that is 30 by 30.
And how am I going to make this box?
I am going to cut corners out of it.
And every corner I am going to cut the same length.
I am going to come in and cut a length of x out.
Ok.
So there I have cut it out.
And now if I was actually making this box, wouldn't I have to fold up the sides.
We are going to have fold lines here, here, here, and here.
And I am going to fold it up.
I am going to take the sides and fold them up, and fold them up, and I am going to get
a box that looks like this.
Badda Bing, Badda Boom.
And just to highlight the bottom where these fold lines are, they are here.
That is not my neatest box ever.
I want to write a formula that represents the volume of this box.
Well volume is length times width times height.
Well, how long is this bottom of this box?
That is this blue line.
Now my box, my cardboard was originally 30 inches wide that I am now going to make this
box out of.
I am going to do that by folding the sides up.
Well that is the length of x here, that is the length of x here, so this blue line is
going to be 30-2x.
Like say that this is 2 inches and this is 2 inches.
Wouldn't the box then be 30, 28, 26 inches.
So I am going to take this cardboard, cut little squares out of it.
That is going to fold.
When I fold it will be a length of 30 minus 2x.
And then how deep is it?
30-2x.
And how tall are the sides of this box?
Well how much did you cut off the corners so you could fold the sides of this cardboard
up? x!
So now I have got flat piece of cardboard with the corners being cut out, being folded
up to make an open top box.
And I can describe the volume of this box based on the fact that I have cut a certain
amount out of each corner, all the same.
And the original piece of cardboard or paper that I am working is 30 by 30 inches.
So before my time runs out, the volume of this box in terms of x is length times width
times height.
You are going to distribute all of this together and get 900-60x-60x+4x^2 times x.
That is going to be 900-120x+4x^2 times x.
And I am not going to finish the last distribution.
I get a formula in terms of x, and these corners that I have cut out that will tell me the
volume of this box that I can make out of this piece of paper that is 30 by 30.
BAM!
Go work on them homework problems:)