# Thread: Need Help with a Problem!

1. ## Need Help with a Problem!

I have a problem for Calculus II....I would greatly appreciate any help.

Suppose that a pyramid has a square base of side 6meters and an altitude of 10 meters. In addition, the density of the pyramid at a point x meters from the vertex is proportional to x (and is given in kg/m^3). In terms of the proportionality constant, what is the total mass of the pyramid?

2. Let b be the side of the pyramid base (6 metres) and h be the height of the pyramid (10 metres).

Any horizontal slice through the pyramid will produce a square with side
b(h-z)/h where z is the height above the bottom.

The density is proportional to the distance x from the apex so:-
density = a times x but note that x = h-z
density= a(h-z)

so the mass of a square at a height z above the bottom is its area (side times side) times the density at that level times dz

dm = [b(h-z)/h]**2 times a(h-z) times dz

Mass is integral from z=0 to z=h of

(a*b**2/h**2) (h-z)**3 dz

Which I assume you can do