Question

A solid has a circular base of radius 3. If every plane cross section perpendicular to the x-axis is an equilateral triangle, then its volume is:
A) 36
B) 12sqrt3
C)18sqrt3
D)24sqrt3
E)36sqrt3

The correct answer was E. Hoping someone could explain how that is.

Answer 1

So we have something like this (drawing):

Each cross section, which is the intersection of a plane and the solid where planes are perpendicular to the base.

If you understand this part, our goal is to figure out the cross section area of any triangle for a given x. Note that the base of any triangle is exactly the length of a chord through the circle! And then once we have cross section area, we integrate:
\( \displaystyle V = \int_{a}^{b} A(x) \ dx \)

If you need more help when you return, please bump this question or tag helpers! :)

Answer 2

A solution using Mathematica is attached.