Question about an 8th grade math problem

[murshid_islam]

TL;DR Summary: It seems like not enough information is given for this 8th grade math problem

For the attached problem,
let b = the side length of the square base of the pyramid and
h = the height of the pyramid
1/3 b²h = 9
b²h = 27

One simple and obvious solution is b = h = 3 (and that's the answer given at the back of the book). But there are infinitely many other solutions here depending on the values of b and h. So why choose b = h = 3? From the image, it kind of looks like the front face of the pyramid is an isoceles right triangle with the base and height being equal. If that's true, then the solution makes sense. What am I missing?

 

[Lnewqban]

Note that only the vertical projection of each side makes an isosceles right triangle with the base and height being equal.
The leaned front faces can't be isosceles right triangles then (slant height > perpendicular-to-base height).

I believe that making the values of b and h of each of the ten wood pieces of the puzzle equal, eliminates a clue regarding their proper orientation.
Note in the picture, that each side accommodates three parts.

 

[murshid_islam]

Note that only the vertical projection of each side makes an isosceles right triangle with the base and height being equal.
The leaned front faces can't be isosceles right triangles then.

I meant the front face looks vertical to me. I don't know if that is actually the case or not.

I believe that making the values of b and h of each of the ten wood pieces of the puzzle equal, eliminates a clue regarding their proper orientation.
Note in the picture, that each side accommodates three parts.

Can you clarify that a bit? I didn't understand what you wrote there.

 

[Lnewqban]

I meant the front face looks vertical to me. I don't know if that is actually the case or not.

What do you call the front face?

Can you clarify that a bit? I didn't understand what you wrote there.

It is just an idea.
Just like you, I believe that the given volume of the pyramid could be achieved with unequal values of b_pyramid and h_pyramid.

For any of those cases, each part would also have to have unequal values of b_part and h_part, which makes me suppose that such characteristic could be used as a clue for choosing the vertical or horizontal orientation of the part.

Judging by the picture, at least all the parts in one level or layer seem to need to measure the same h_part.

 

[murshid_islam]

What do you call the front face?

This face:

For any of those cases, each part would also have to have unequal values of b_part and h_part, which makes me suppose that such characteristic could be used as a clue for choosing the vertical or horizontal orientation of the part.

Judging by the picture, at least all the parts in one level or layer seem to need to measure the same h_part.

What do you mean by "part"? Do you mean the faces? Or do you mean the 10 smaller pieces the puzzle consists of?

 

[Lnewqban]

This face:

View attachment 357424

Then, that face can’t be vertical.
Note that the height of that face is called slant in the diagram of post #2.
Also note that it is the perpendicular height the “h” computed in the shown equation of the volume.

What do you mean by "part"? Do you mean the faces? Or do you mean the 10 smaller pieces the puzzle consists of?

I meant each of the 10 smaller pieces the puzzle consists of.

 

[BvU]

I'm asking myself: have I seen use of the given information 'comes with the pieces arranged in a cube' ?
There must be lots of 90 degree angles on the pieces ...

8th grade ?

$\$

 

[murshid_islam]

Then, that face can’t be vertical.
Note that the height of that face is called slant in the diagram of post #2.
Also note that it is the perpendicular height the “h” computed in the shown equation of the volume.

I'm not getting why a pyramid with a square base cannot have its apex directly vertically above one of the corners of the square. Can you explain?

I'm asking myself: have I seen use of the given information 'comes with the pieces arranged in a cube' ?

I can't think of a use of that information.

8th grade ?

Yes, the problem is from the book "MYP Mathematics 3: A Concept-Based Approach" by David Weber, Talei Kunkel, Rose Harrison and Fatima Remtulla

 

[Lnewqban]

I'm not getting why a pyramid with a square base cannot have its apex directly vertically above one of the corners of the square. Can you explain?

It can.
I may be wrong, but I have assumed that the picture shows the apex centered on the base.
Look at the apex single piece.

Copied from:
https://en.m.wikipedia.org/wiki/Square_pyramid

“If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid,...”

 

[murshid_islam]

It can.
I may be wrong, but I have assumed that the picture shows the apex centered on the base.
Look at the apex single piece.

Copied from:
https://en.m.wikipedia.org/wiki/Square_pyramid

“If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid,...”

Since the problem did not specify it's a right square pyramid, I assumed it doesn't necessarily have to be one and that it could be an oblique square pyramid.

 

[Lnewqban]

Since the problem did not specify it's a right square pyramid, I assumed it doesn't necessarily have to be one and that it could be an oblique square pyramid.

Perfectly possible.

 

[SammyS]

This face:

What do you mean by "part"? Do you mean the faces? Or do you mean the 10 smaller pieces the puzzle consists of?

Yes, I agree. That face looks to be vertical. Also, the face that's viewable in that same image, and which is adjacent to indicated face also looks to me to be vertical. These two faces share a common edge, which is the altitude of the pyramid. Not only are these two faces perpendicular to each other, they both are perpendicular to the square base and furthermore, each is perpendicular to the oblique face it intersects along an edge. It seems to me that having all these surfaces with these perpendicular relationships is needed because, the same pieces go together to form a cube.

Up to this point, there is no reason to set $h=b=3\,\rm{cm}$. But having $h=b=3\,\rm{cm}$ makes the numbers work out nicely. However, there is an additional interesting feature of such a square oblique pyramid for which $h=b$. If we take three such pyramid's, we can place them together in such a way that they form a cube. This cube has a volume of $h^3$ or equivalently $b^3$ or in or our case, a volume of $3^3=27\,\rm{cm}^3$.

That's some 8th grade math problem !

 

[pbuk]

One simple and obvious solution is b = h = 3 (and that's the answer given at the back of the book).

The answer in the book is wrong.

In order to tessalate to form both a cube and a pyramid it is necessary to rotate the pieces relative to each other and so the triangular faces of the pieces must have rotational symmetry. They must therefore be equilateral triangles, and so therefore are the triangular faces of the pyramid.

After two applications of Pythagoras we find that $b = 3\sqrt[6]{2} \approx 3.37\text{ cm}, \, h = \frac{\sqrt{2}}{2} b \approx 2.38\text{ cm}$

Edit: for a better image see https://www.wickeduncle.co.uk/childrens-presents/great-pyramid-of-giza-puzzle

 

[murshid_islam]

In order to tessalate to form both a cube and a pyramid it is necessary to rotate the pieces relative to each other and so the triangular faces of the pieces must have rotational symmetry. They must therefore be equilateral triangles, and so therefore are the triangular faces of the pyramid.

I did not follow that. Can you elaborate a little?
Also, is a regular 8th grader supposed to know that?

 

[SammyS]

I did not follow that. Can you elaborate a little?
Also, is a regular 8th grader supposed to know that?

I agree with you. That explanation is well beyond 8th grade level.

I doubt that the puzzle in the link can be made into a cube.

 

[kickaxe]

A square pyramid is always 1/3
the size of the cube.

Everyone making it so complicated lol.

 

[kickaxe]

And since the cube is clearly 3 on a side, put that in the formula for a pyramid. Or just put 7

 

[kickaxe]

There are not infinite answers, there are clear equations given.

 

[WWGD]

And since the cube is clearly 3 on a side, put that in the formula for a pyramid. Or just put 7

If the cube has side 3, its area is $6L^2=54$. Then $(1/3)54=18$.

 

[kickaxe]

They asked for the dimensions and volume not the area. Read the OP

Otherwise correct :)

 

[murshid_islam]

A square pyramid is always 1/3 the size of the cube.

I'm not getting how that helps. The problem statement says that 1 cube is to be rearranged into 1 square pyramid.

And since the cube is clearly 3 on a side, put that in the formula for a pyramid. Or just put 7

The cube has a volume of 9 cm³