Help with volume by cross section question

[student93]

Homework Statement

See the problem attached in this post.

Homework Equations

See the problem attached in this post.

The Attempt at a Solution

I set my limits of integration with respect to z axis and got an upper limit of 2 since that's the vertex point/height of the pyramid and my lower limit as 0 since the lowest possible point in regards to the z axis is 0. The area of a square is s^2 and I set my integrand as ∫s^2 dz, from 0 to 2 and got 8/3, which is actually the correct answer. However, this is just a coincidence since I just realized you can't take the integral of s^2 with respect to the z axis (It's necessary to convert s into some term of z). How exactly do I convert s into a term of z so that I can set up the correct integrand?

 

[SteamKing]

Sorry, I see no attachment to your post.

 

[student93]

Sorry, I see no attachment to your post.

I edited the post with the problem.

 

[SteamKing]

I assume that 's' represents the length of the side of the base of the pyramid.

The calculation of 's' depends on knowing the dimensions of the base, i.e., the location of the endpoints of the sides. If you draw lines connecting the endpoints of the base with the vertex or apex of the pyramid (the tippy top point), how would you determine the dimensions of any intermediate cross-sections of the pyramid? Is there some sort of formula involving z which could be used?

 

[student93]

I assume that 's' represents the length of the side of the base of the pyramid.

The calculation of 's' depends on knowing the dimensions of the base, i.e., the location of the endpoints of the sides. If you draw lines connecting the endpoints of the base with the vertex or apex of the pyramid (the tippy top point), how would you determine the dimensions of any intermediate cross-sections of the pyramid? Is there some sort of formula involving z which could be used?

You could use Pythagorean Theorem to get (1)^2 +(2)^2 = 5, thus the hypotenuse connected from the vertex to the x-axis would be √5, however I don't know where to go from there.

 

[SteamKing]

You're thinking from a numerical point of view. Look at the geometry of the pyramid. Let's say the length of the base side is 's' and the pyramid has a height of 'h' from the apex perpendicular to the base. What is the length of the cross-section side at z = h/2? At z = h/4? At z = 3h/4? There is a simple relationship for all of these values.

 

[student93]

You're thinking from a numerical point of view. Look at the geometry of the pyramid. Let's say the length of the base side is 's' and the pyramid has a height of 'h' from the apex perpendicular to the base. What is the length of the cross-section side at z = h/2? At z = h/4? At z = 3h/4? There is a simple relationship for all of these values.

By cross section you mean a square right? So wouldn't the length of the cross section just be the same value as z? For example if z=h/2 then the length of the cross section side also equals h/2 since all four sides of a square are equal?

 

[student93]

I ended up using the similar triangles theorem and ended up getting s=2-z, and solved for the integral and got 8/3 which is the correct answer. I'm assuming I used the correct method this time around?

 

[SteamKing]

The key is understanding that the length of the side of a pyramid at any altitude is proportional to the distance of the cross section from the apex of the pyramid. At the apex, s = 0 obviously, and at z = h, the length of the side s = 1 say. Then, at z = h/2, s = 1/2, and so on.