Minecraft and Calculus – Optimization of Volume

I’m holding 144 diamond blocks that I want to use to construct a very large house for myself because I am the king of this village and the King deserves the biggest possible house now what I say the biggest I don’t mean the tallest and I

Don’t mean the thickest I mean it has to have the biggest possible volume on the inside because I have a bunch of these paintings and item frames that I want to place in it and I need the most amount of space now to do that I could

Obviously copy one of these designs or I could do a little bit of math and figure it out using calculus and quadratic equations welcome to what the math today we’re going to be using minecraft to do a little bit of mathematics and so let’s start with a little bit of math here and

We’re gonna start with assuming that we’ll have 144 diamond blocks it doesn’t have to be diamond it can be anything it doesn’t have to be 144 but I chose this number because it’s a little bit easier to calculate and we’re trying to basically build ourselves a house or

Just say it’s going to be just a rectangular house with no roof or anything like that that has the highest possible volume on the inside so we’re gonna write that right here maximum volume out of those 144 blocks we have to try to figure this

Out so one of the easier ways of doing this is to make an assumption that let’s just say we’re going to have on the bottom here which is going to have a square in other words our foundation or our base is going to be square base and

It’s a square that has a set number of squares on both sides so both of these are going to be both of these sides are going to be equal so and then we’re obviously gonna have some server height here that will produce walls for us and

We’ll then turn this into a house so in other words we actually have several different variables here we have our variable X which is going to be our height so X here is going to represent height and then we have our variable Y right here which is going to represent

The length and the width of the house now it might not be the best to have a square foundation or square base but for this the simplicity sakes we’re going to start with a square foundation so here lengthen with our Y so this is y this is

Y and this right here is X and to start our calculation let’s just assume that we’re going to place our 144 blocks in a square shape just so we can start with a foundation in other words we’re going to have 144 squares lying like this in the

Square form and so here we go this is what we get if we lay them out like this and this is basically 12 by 12 because this is actually why I picked 144 because 12 times 12 is 144 so basically you can find this by taking a square

Root of 144 and you get this sort of a shape all right so that’s a good start so this is actually a hundred forty-four diamonds or a diamond blocks and to turn this into a house we’re going to cut out these little squares like this and what

We’re going to do next is basically build a box we’re essentially just building the box out of this so we’re going to cut these out and then fold right here to make walls so right here we’re gonna have these little folds to turn this into a box in other words

Here’s one example I’m going to take out squares of a size of about three by three and this will give me a shape that looks like this now these parts right here are going to be our wall so basically just going to fold them up now

And place them on on top of this now now remember this is actually upside down so this part will be actually our roof once we flip it over and because we don’t really need the floor right we can live without it so we’re just going to have

Sand as our floor and then this will be our roof and these are actually our walls and so here’s one of them that has been just folded over and it gives us a wall that’s about six long and three high so if we were to do this for all of

These it would give me a shape like this and now obviously once again this is upside down so all I have to do now is remove these and turn them into a roof but basically this is what our first creation is like but is this the maximum

Volume will I actually have the most amount of space for me to place my paintings here so first our place in paintings will this be the maximum volume and unfortunately the answer is not this is not the highest volume so in other words I can build a house that

Will allow me to put even more paintings by changing the dimensions a little bit so let’s find out what would be the perfect size here we’re gonna do this by graphing first and then I’m gonna show you how to do this manually using calculus now let’s start with the

Equation that we can use first and so here we’re gonna be talking about these little squares that we cut out and we we know that the height as we mentioned before is actually X so both of these are X the total length of that one side

If you remember this was actually 12 now what we can do is now we can rewrite this whole side as this we can rewrite this as 12 minus 2x so because this is 12 and this is X and X this this whole side and this side right here that I’m

Talking about this side will be 12 minus 2x in a similar fashion this other side will also be 12 minus 2x because we’re actually using a square so we can now write side 2 is also 12 minus 2x and side number 3 is our height and if you

Remember the height here is just X now what we need to find volume oh that’s easy side 1 times side two times size 3 and obviously this will give us I’ll just write this in a somewhat easier way to read so it’s going to be given give

Us height x first side x the second side and that’s volume and we can actually rewrite this now in a slightly easier to read format of X multiplied by 12 minus 2x and all of this is squared and this right here is our volume for this shape now we’re looking for the highest

Possible volume with a certain X now how do we do this well the easiest way is to obviously graph it and to graph this I’m going to be using decimals decimals calculator it’s absolute frickin fine alone decimals calm and here I’m going to launch it and let’s

Just type this in so X multiplied by 12 minus 2x and we’re going to have squares on the outside and here we go so this is our function and all we need to do now is I just remove this for a second and we’re basically looking at a graph that

Is known as a cubic graph and this graph has a very interesting shape it’s sort of like an s-shaped and specifically in this particular graph we’re looking for what’s called a a local maximum so what you need to do is you need to scroll up until you find this right here this

Little peak right here because this part will not never actually end and this other part will also never ends the part on the bottom is not really important because it just shows us when volume is zero but we’re actually looking for the highest volume and the highest volume is

At this point right here because this this represents volume X here represents the the height the acts that we used before and so if you click on this it will give you the answer and the answer here is 2 and the volume that you’ll

Have at the value of 2 is 128 so if I write 2 here this is 2 and this is 2 then this will give us the highest possible volume and and this of course is 12 minus 2 times 2 so this would be

12 minus 4 or 8 and this will also be 8 so what we’re looking to create here is a house with the height of 2 and the side of 8 and 8 and so this will give you the perfect volume for this particular amount of blocks and here we

Go this is what it would actually look like and is unfortunately no door so we may have to make one the inside of this little house has volume of 128 cubes or I guess minecraft squares mycroft cubes and look at spiders and we can basically place as many paintings now as we want

This will give us highest volume and on top of that because we cut out those little squares because we actually cut out four four little squares four little squares from each side if this also means that we now have 16 more diamond blocks to spare and to use for something

Else we can actually maybe build a barn or possibly use these for some other interesting construction now this is how you would do this using graphing but let me show you also how to do this using just algebra and calculus basically calculating it manually using your hands

And to do this manually we’re going to go back to this formula right here and we’re now going to expand it by using foiling and we’re going to basically cross multiply these two brackets what this will give us is this so here if you remember how to foil and I’m gonna do

Relatively quickly but if you don’t remember what one of the previous videos I posted on how to foil so here multiplying these things four times I’m going to cross multiply them to get the following this will give us X multiplied by 144 minus 48 X and then plus 4x

Squared and lastly let’s multiply everything by that X outside and this will give us 144 x minus 48 X square plus 4 X cube so this is what we have this is the actual function and this is what we graphed before now what is calculus and what is it horrible

Calculus in essence is to basically find rates of changes or or how fast things change in relation to something else now here what we want to do is this we want to find this point right here when things are no longer changing they’re actually at 0 in other words at this

Point right here things are stationary this is called a stationary point because on this point for example we can see that it’s still kind of increasing here it’s decreasing but here it’s absolutely zero so this is actually when nothing is changing so what we want to

Find out is we want to find that one point when nothing changes anymore so we’re going to be looking for the stationary point and to do this we need to find what’s called a derivative or what we’re doing is with differentiating this function we’re going to call it Y Prime and this

Is how this works so I’m gonna write this little rule here for calculus this is a calculus rule and the rule works like this for every value we’re going to put our exponent original exponent in front of the number I’m going to then take our X and reduces exponent by one

Now this is a calculus rule that you will either study in high school or has have already studied and possibly forgot and anyway so this is the rule it’s relatively simple so basically what this means is the exponent here is 1 the exponent here is 2 the exponent here is

3 so we’re going to put these exponents in front first so here it’s just going to be 1 and then we’re going to reduce this exponent by 1 so this will turn into 0 this will turn into 1 and this will turn into 2 and so what this

Function will now become is this it’s going to become 144 minus 2 times 48 is 96 minus 96 and then it’s going to lose its 1 power and become just X and finally 3 goes in front becomes 12 plus 12 and then it loses its power becomes x

Squared now this is what we would call in calculus a differentiated function or a derivative of a function what does this represent well this represents the actual graph for rates of change now we don’t really need to graph this all we need to do now is find this one

Stationary point and like I said before stationary point is when the actual rate of change is zero so we’re going to turn this into this it is going to be equal to zero so I’m going to rewrite this right below 12 X square minus 96 X plus

144 equals to 0 and this is a quadratic function this is something you may have done in grade 8 or grade 9 and to solve this we can do two things one of those things is factory and one of those things is using a quadratic formula which I’m

Going to do right now let’s actually use this little spot right here to write the quadratic formula and the quadratic formula goes like this x equals and there’s going to be always two values of X for quadratic formulas it equals to minus B and B here is this middle number

This is B and it’s minus B so in this case because this is already minus is going to be just 96 plus minus square root of b square minus 4ac this is a and this is C so that’s kind of like how we try to solve these quadratic formulas

And then all of this is what would be divided by 2a so let me just erase some of the stuff so we can actually do this now to make this even more simpler I can actually divide everything by 12 here and this is something we can do because we’re basically dividing everything by

12 including zero to get this we’re gonna get x squared minus 8x plus 12 equals to zero and so let’s use the quadratic function now and it’s going to be minus B so that’s eight plus minus and that’s because we’re gonna get two values B square is 64 4ac is 4 times 1

Which is 4 times is this which is 12 in other words it’s minus 12 times 4 which is 48 and then all of this divided by 2 and so now if we actually try to simplify this middle part we’ll get 8 plus minus square root of 16 divided by

2 and this can be now written like this X 1 this is our first value is going to be 8 plus square root of 16 is 4 8 plus 4 divided by 2 and X 2 this is the minus part will be 8 minus 4 divided by 2 so

First value is 8 plus 4 is 12 divided by 2 is 6 second value is 8 minus 4 is 4 divided by 2 is 2 now so we have two answers 6 & 2 now we have to go back to the original picture and try to see if

Any of these values don’t fit so here this was how we found one of the sides if this side is sick and we’ll put six in here this will actually give us zero and it cannot be zero so we cannot use this value because it will give us zero but two will not

Give us zero – oh actually give us eight so and that’s what the answer is so the answer is two four for the height and eight for the sides and so basically this is how you would solve this using calculus using quadratic equations and hopefully this was kind of clear

In other words next time you’re building something in games like Minecraft and you want to optimize your volume and find the best possible volume for a certain shape you may want to do some calculation in the next video or one of the future videos we’re going to do the

Same in a game called from the depths and I’m gonna show you how to construct an optimally sized ship as well using a very similar principle to this anyway hopefully you guys enjoyed this thank you so much for watching and gave me later I’m going to try to make my house

A little bit more comforting and a little bit more beautiful thank you and bye bye