# 2 cubes infused together

For the following question:

I had absolutely no clue how to approach it, and even looking at the solution I struggled (other smc q25s I can sort of understand/attempt but this one I especially had trouble with).

I was ok with the first half until I had to work out h.
I managed to complete it using cosine rule, but the ms just does it using 3D Pythagoras in one line whereas I took like 20 .

I dont know why but I cant visualise doing 3D pythagoras here (or even If I can, the ms just breezes past it in one line, which is telling me this is the easy part and I dont really know how)

If anyone could help out, it would be greatly appreciated.
(edited 3 weeks ago)
Working:
The y-y-y base equilateral triangle has an altitide sqrt(3)y/2 and the triangle centre (point under the top vertex of the tetranedron) is 2/3 of the way along that so sqrt(6). Then its pythagoras on the h-sqrt(6)-3 right triangle.

Or slightly more direct to get the sqrt(6), the base triangle centre forms a 30-60-90 where the side opp the 60 is y/2, so the hyp is (2/sqrt(3))*(y/2) so sqrt(6)
(edited 3 weeks ago)
Ok thank you that makes a lot more sense.

I had to draw about 10 diagrams to visually understand this question but I think I get it now.

Also just a bonus q, would you say its worth memorising the areas for more regular shapes?
I learnt herons formula and area of equilateral triangle, but should I try memorising the areas of pentagon, hex... to save time?
Or are there any other formulas that you think are worth learning?
Original post by mosaurlodon
Ok thank you that makes a lot more sense.
I had to draw about 10 diagrams to visually understand this question but I think I get it now.
Also just a bonus q, would you say its worth memorising the areas for more regular shapes?
I learnt herons formula and area of equilateral triangle, but should I try memorising the areas of pentagon, hex... to save time?
Or are there any other formulas that you think are worth learning?

Its honestly less about learning new formula. I doubt theyd do a question based on heron if there wasnt an elementary way to do it. Similarly for the equilateral as here its just 1/2 sin(60) x^2 and you should know sin(60) as a surd. You often do the more complex shapes by splitting them up into simpler shapes (triangles?) so a 12 sided regular polygon has an angle at the center of 30 ...

A couple of things from this question

be confident with your 1:sqrt(3):2 side ratios for a 30-60-90 triangle (or the equivalent variants 1/2:sqrt(3)/2:1 and 1/sqrt(3):1:2/sqrt(3)). The latter was used here. Its basically just trig, but often ukmt questions are based on side ratios and similar triangles.

Understand how an altitude is used to go from 60-60-60 to 30-60-90 and where the simple triangle center is and how its used to derive cos and sin rules and get the height of a point above the base of a semicircle (similar to thales).

Get in the habit of splitting up triangles (or more complex shapes into triangles). So drop altitudes, connect tangent points to centres, ... as they construct right triangles.

(edited 3 weeks ago)
Original post by mosaurlodon
Ok thank you that makes a lot more sense.
I had to draw about 10 diagrams to visually understand this question but I think I get it now.
Also just a bonus q, would you say its worth memorising the areas for more regular shapes?
I learnt herons formula and area of equilateral triangle, but should I try memorising the areas of pentagon, hex... to save time?
Or are there any other formulas that you think are worth learning?

Its worth noting that this ms significantly overcomplicates it. If you took the base as (one of) the 3-3-? 45-45-90 triangle then its area is trivially 9/2. Then the perpendicular height is 3 and the volume of the cone is therefore 9/2.

A bit like the above tips, its not uncommon for questions to be asked about the base*height triangle area formula (or similar volume), but the "base" is not the side on the bottom. So its more about thinking how you apply basic formulae "intelligently" rather than learning new ones.
(edited 3 weeks ago)
Oh yeah wait what?? That is soooo much easier.

man this question would take less than a minute to solve if someone spotted that, but obvs its q25 so spotting that mustve been hard.

Thanks for your tips ill keep that in mind - I guess I just have to work on becoming more intelligent to apply the formulas correctly.
Original post by mosaurlodon
Oh yeah wait what?? That is soooo much easier.
man this question would take less than a minute to solve if someone spotted that, but obvs its q25 so spotting that mustve been hard.
Thanks for your tips ill keep that in mind - I guess I just have to work on becoming more intelligent to apply the formulas correctly.

Must admit, Im surprised at the ukmt ms in that they overcomplicate the solution as the question/solution will have been checked by a few people and doing it the simpler way tends to make it less of a "q25" and almost doable in your head. Having said that, most of the smc questions (including the harder ones) should be able to be done in a few lines of scribbling (with a decent sketch) and doing a full solution like youd do in gcse/a level often just eats time. Also, the above ways to get the sqrt(6) do crop up in other ukmt questions so its not bad practice.

For some of the earlier questions though, the ms can be a bit laborious compared to a quicker problem solving approach and its worth practicing the latter both to speed up and also give you another way of approaching a problem. For this question, the approach is the same, its just that
(edited 3 weeks ago)
Original post by mosaurlodon
Oh yeah wait what?? That is soooo much easier.
man this question would take less than a minute to solve if someone spotted that, but obvs its q25 so spotting that mustve been hard.
Thanks for your tips ill keep that in mind - I guess I just have to work on becoming more intelligent to apply the formulas correctly.

As a wrap up and partially about some "reading" which may help, getting the volumes of spheres, tetrahedron, cones can be traced back to archimedes though the technique about showing that volumes are the same was rediscovered/renamed by cavalieri. Archimedes proved the famous
cone+hemisphere = cylinder
https://thatsmaths.com/2019/11/28/archimedes-and-the-volume-of-a-sphere/
by taking horizontal slices and showing they had the same volumes which obv is similar to integration, though historically you showed that things had the same area/volume, rather than getting an expression. You can do an equivalent argument with the tetrahedron in this question to show
6 tetrahedron = 3 square based pyramids = cube
so the tetrahedron volume here is 27/6 ....

For ukmt, its less about memorising the final formulae (though you should know all these) and more about thinking about how the arguments are constructed (so decompose areas/volumes into more elementary shapes ...) and in this question as soon as you drew the y-y-y triangle which bisected the overlapping diamond into two tetrahedron, you could write down
4^3+3^3-2*3^3/6
(edited 3 weeks ago)
man khan academy is so useful for this stuff.

tbh, I didnt even know about cavalieri, I dont think its taught at school (or atleast I dont remember learning it), but working through the stuff you sent, makes it a lot more obvious and basically common sense
Original post by mosaurlodon
man khan academy is so useful for this stuff.
tbh, I didnt even know about cavalieri, I dont think its taught at school (or atleast I dont remember learning it), but working through the stuff you sent, makes it a lot more obvious and basically common sense

Its usually worth reading around (and/or asking about) the gcse syllabus if/when you do a slightly unusual question as there is a fair amount of historical stuff on there which is elementary/can be solved using gcse techniques, but may appear somewhat unusual. So things like

dots (difference of two squares) and am/gm inequality is related to babylonian multiplication where you had tables of square numbers and used them to compute the multiplication of two numbers.

pythagorean triples pop up a lot and the classic euclid proof using dots and parity arguments. Similarly for variants on the hcf method

thales (similar triangles, right triangle in a semicircle) is a foundation stone of geometry

altitudes can split isosceles triangles into right triangles (so your original method where you used the cos rule could be replaced by a single bit of simple trig if youd split the 30-120-30 into two 30-60-90 triangles). Trig appears a bit on ukmt, but side ratios/similar triangles seems to be the way to do most such questions efficiently

divisibility, remainders primes, ....

Theres little on there that cant be approached using gcse knowledge, but practice/reviewing the basic ideas will make it much more familiar. Coming back to your point about cavalieri, the archimedes cone+hemisphere=cylinder is one way to justify the sphere volume formula using gcse (pythagoras) (or using a "flat hedgehog" sheet of thin cones) but it isnt on the syllabus as such. Its more about understanding the ideas (southall's yes but why book), rather than just learning the formula.
(edited 3 weeks ago)