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So here is our GeoGebra solution to example 1. This purple region represents all of the
possible solutions to the problem and we have found the intersection points A, E, D and
O and tested our objective function for those 4 points. We found the maximum was 5 at E
and now we want to do a little bit more exploring with the objective function. The first thing
we need to know are what are possible values. We have some possible values here. Our minimum
is 0 and for sure the function cannot below 0 because it is a real problem. We see that
our maximum is 5, but let us go a little bit above that – let us say 10. We see that
our numbers have one decimal point. So we are going to let our increment be 0.1. We
go to our slider tool and we put it somewhere in the drawing pad. We are going to call it
Volume. The reason we don’t call it V is that sometimes it may be C for Calories and
we might already have a point by that name. So we might as well call it by its whole name.
We said a minimum of 0, maximum of 10, that increment is good 0.1. I always make my sliders
a little bit longer – 200. Click on Apply so that I have more control over the values.
Now this is the value of the objective function. Now we need the line for the objective function.
We click down here in the input bar and type in our objective function. So it is 0.4*x+0.6*y
equals – and then we want it to be this value of the slider so we type in the slider
name “volume”. And we hit enter. So this line represents all of the points in the whole
world that give this volume by the objective function.
We are only interested in points in the purple shaded region. So let us make a point on this
line so that we can see its values. We go to the new point tool, make the line glow
and click on it. So F is always a point on that line. Let us get to show its coordinates
and not just its name. Double-click, go to Properties and on the Basics tab, we will
tell it to show its value. The value is the coordinates. Let us test to make sure that
this is working so as we move it – there it goes.
And now if we want to, we can create a nice looking text that shows how this point gives
that volume. So we go to the insert text tool, we click and now – having done this 100
times – it is much better to go and copy the formula. So we are going to our text,
and there is our formula already marked. We hit Ctrl+C, come back, hit Ctrl+V. Now there
are 2 things we need to check. Make sure all of the double quotes are standard double quotes
and not smart quotes from Word. And the 2nd thing is that we have to put in place of A
our coefficient 0.4, in place of P the name of our point which was F, in place of B our
coefficient 0.6, in place of P, again we have F and this IS the name of our slider so we
are done. And now we hit Latex formula. If you don’t you can double-click and come
back and fix it. So we are just going to leave it as if we have forgotten it. We can see
that it is giving us junk so double-click and now click LaTeX and then Ok. It looks
exactly right. If we want to make it bigger, we right-click,
choose Properties and come up here and make it 18. Now while we are in here, let us change
the style of the line to be dash-dot and thicker, the style of the slider – where is the slider
– Look for numbers, sliders are numbers and make it dash-dot and thicker. Where is
our point. It is F. We definitely do not like that color. We are going to make it purple
and we are going to make our text purple so it matches. There we go. We are done. Now
let us check the coordinates are dynamic so right now we have 2.08 and 2.62 match the
coordinates of F. Let us move F and the coordinates change. So everything is as it should be.
Now with this slider and the line and the point on the line, we can check the value
of the objective function for every point in the purple shaded region.