Tip:
Highlight text to annotate it
X
This was our solution to example 1. We can consider any point in this purple area a possible
solution to our problem. The question is: Why is E the point where there is a maximum.
That is the question. Supposedly only the point E gives me a volume of 5. Let us make
a slider with the volume and then see which points in this purple area give us that volume.
So we select our slider tool and click somewhere. We will name our slider “Volume”. It has
to be positive so minimum is 0. But let us allow it to be bigger than 5. We will let
it go to 10. We will use the increment 0.1 since that is the increment that we see here.
Let us make it a little bit wider so we have a little more control. We let it be 200. And
there is our slider. Let us move it to 2. Which points in here give us a volume of 2?
How did we get this volume of 4.2? We multiplied 0.4*x and 0.6*y. That was our objective function.
So we click down here in the input bar and we type in our objective function. 0.4*x +
0.6*y and we say that we want it to equal that number there which is Volume. That is
our variable. Right here – all of these points in the
purple area on this line give us a volume of 2. Let see what these points are. We select
our New point tool and we make the line glow and now F is “connected” to the line.
So if we click and drag F it will not move off the line. Now we want to see its coordinates
and decorate a little bit. So all of these objects are now connected. So what we are
saying is that – if we have the point (2.03, 1.98), it will give us a volume of 2. So supposedly
if we move the slider to 4.2, this line should pass through A. Let us see if that happens.
It passes through A and then all of these points also give a volume of 4.2 So we can
go along here and find other points. There is (6,3) also gives us a volume of 4.2. If
we slide to 4.8 we should get D. There are not very many points there in the purple region.
We are getting less and less points. What happens if we slide out to 5? We get only
the one point in the purple region and that is E.
Now the reason that is, is because this purple region is what we call “convex”. Convex
means that the point stick out but not the are itself. The intersection points stick
out but we don’t have any “star points” where area is sticking out. And then, because
there is no areas sticking out, when the line gets outside it only has one point that makes
that volume. And what happens if we go bigger than 5. Oops – there are no points in the
purple area. There are no points in the purple area which give a volume of 5.5. There is
only the one point that gives a volume of 5. And then there are lots of points that
give smaller volume. This is a way to see “how and why and if” checking intersection
points will work.