For each pair of solids, determine if their volumes are the same or different. If the volumes
are different, identify the solid with the greatest volume. Explain your reasoning.
a. A prism and a pyramid have the same height. The pyramid’s base has 3 times the area of
the prism's base.
b. A pyramid and a cylinder have bases with the same area. The cylinder’s height is 3 times
that of the pyramid.
c. A cone and a cylinder have the same height. The cone’s radius is 3 times the length of the
cylinder’s radius.
a. The volume of a prism is given by V = Bh, where B is the area of the base and h is the height. The volume of a pyramid is given by V = (1/3)Bh, where B is the area of the base and h is the height. In this case, both solids have the same height, so the volume of the pyramid is 1/3 of the volume of the prism. Therefore, the volume of the prism is greater than the volume of the pyramid.
b. The volume of a pyramid is V = (1/3)Bh, where B is the area of the base and h is the height. The volume of a cylinder is V = πr^2h, where r is the radius and h is the height. In this case, the bases have the same area, so the volume of the cylinder is greater than the volume of the pyramid since the height of the cylinder is 3 times that of the pyramid.
c. The volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height. The volume of a cylinder is V = πr^2h, where r is the radius and h is the height. Since the height is the same for both solids, the volume of the cone is 1/3 of the volume of the cylinder. Therefore, the volume of the cylinder is greater than the volume of the cone.
make all of your answers about half a sentence
a. The volume of the prism is greater than the volume of the pyramid because the base of the pyramid is 1/3 the area of the base of the prism.
b. The volume of the cylinder is greater than the volume of the pyramid since the cylinder's height is 3 times that of the pyramid.
c. The volume of the cylinder is greater than the volume of the cone because the cone's volume is 1/3 of the cylinder's volume.