Question 1084033
If option C is a square base pyramid with base sides measuring 36 inches,
that is definitely larger.
 
If option C is a square base pyramid with base area measuring 36 square inches,
that is smaller than all the other options.
 
The sphere (D) and the square base prism (B) are larger than the popcorn container.
 
Here is a side view of the three items:
{{{drawing(675,300,-6,21,-6,6,
circle(0,0,4.5),arrow(0,0,0,4.5),
arrow(0,4.5,0,0), locate(0.1,3,4.5in),
line(6,-5,12,-5),line(7,5,11,5),
line(6,-5,7,5),line(12,-5,11,5),
rectangle(14,-5,20,5),locate(8.5,-5,6in),
locate(8.5,5,4in),locate(16.5,-5,6in),
locate(14.1,0.5,10in),locate(-1,-2,shpere),
locate(16,3,prism)
)}}}
 
It is obvious that the popcorn container fits inside the prism.
 
The comparison with the sphere is not so obvious.
 
The shape of the popcorn container is called a frustum of a pyramid.
That is the "stump" left when we cut off a pyramid parallel to its base.
Imagine a pyramid square bottom base with side length of 6 in and a height of 30 inches.
Its volume is {{{(1/3)(base_area)(height)=(1/3)(6in)(6in)(30in)=6*6*10}}}{{{in^3}}} .
Now cut it at a height of 10 inches.
The part above the cut is a smaller pyramid {{{2/3}}} scale replica of the original pyramid.
Its volume is {{{(2/3)^3=8/27}}} times the volume of the original pyramid.
The bottom part is your frustum.
{{{drawing(125,400,-6,4,-1,31,
triangle(-3,0,3,0,0,30),line(-6,0,4,0),
red(line(-5.8,10,3.5,10)),arrow(-4,21,-4,30),
arrow(-4,19,-4,10),locate(-4.8,20.6,20in),
red(arrow(-5.5,0,-5.5,10)),locate(-5.3,6,10in),
red(arrow(-5.5,10,-5.5,0)),locate(-1.2,1.3,6in),
locate(-1.2,10,4in),green(arrow(0.2,30,-7,30))
)}}} The volume of the frustum is {{{1-8/27=19/27}}} times the volume of the original pyramid.
It is {{{(19/27)(6*6*10)}}}{{{in^3=19*6*6*10/27}}}{{{in^3=19*4*10/3}}}{{{in^3=about253.33}}}{{{in^3}}} .
 
The volume of the sphere is
{{{(4/3)(radius)^3pi=4*(4.5in)^3*pi/3=about381.7}}}{{{in^3}}} .