The base of a solid is the circle x^2 + y^2 = 9. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

Question

36√3
36
18√3
18

Answer 1

So the cross sections (parallel to the base) are also circles, of formula
x^2+y^2=r^2, where r is radius.
Now given the vertical cross sections are equilaterial, that means the sides at x or y at zero, is equal to twice the radius of the base, or 6.
so the height of the solid is 6/sqrt3 (sketch a equilateral triangle of side 6), or h=2.44
at this point, you dont need calculus and integration (you can do that), but knowing the volume of a cone is 1/3 base area * height, or
volume= 1/2 (PI*3^2)(2.44)

Answer 2

idk