Question

What can be concluded about a cylinder and a cone with the same height and radius measures? Check all that apply.

A. The cone has the greater volume.
B. The cone and cylinder have the same base area.
C. The volume of the cone is 1/3 the volume of the cylinder.
D. The cylinder can fit inside the cone.
E. The cone and cylinder have the same diameter.

Answer 1

Answer:

$\text{volume of cone=}\frac{1}{3}\text{volume of cylinder}$

Step-by-step explanation:

We have to choose the correct statement about the volume of cone and cylinder with same height and radius.

Let r and h be the radius and height of cylinder and cone

$\text{Volume of cone=}\frac{1}{3}\pi r^2h$ ....(1)

$\text{Volume of cylinder=}\pi r^2 h$   ...... (2)

$\frac{\text{Volume of cone}}{\text{Volume of cylinder}}=\frac{\frac{1}{3}\pi r^2h}{\pi r^2 h}$

$\frac{\text{Volume of cone}}{\text{Volume of cylinder}}=\frac{1}{3}$

$\text{volume of cone=}\frac{1}{3}\text{volume of cylinder}$

Option C is correct

Answer 2

Cone with same dimensions as cylinder is less space occupying. The volume of such cone is one-third of the volume of same dimensional cylinder. The cone and cylinder specified have same diameter.

What is the volume, base area and diameter of a cone with radius r and height h?

$\rm Diameter(\text{Of base}) = 2 \times radius = 2r \: \: units\\\\Volume \: of \: cone = \dfrac{1}{3}\pi r^2h \: \: unit^3\\\\Base's \: area = \pi r^2 \: \: unit^2 \:$

What is the volume, base area and diameter of a cylinder with radius r and height h?

$\rm Diameter = 2 \times radius = 2r \: \: units\\\\Volume \: of \: cone = \pi r^2h \: \: unit^3\\\\Base's \: \: area = \pi r^2 \: \: unit^2 \:$

Thus, it can be seen that:

The cone and cylinder have the same base area.

The volume of the cone is 1/3 the volume of the cylinder.

The cone and cylinder have the same diameter but only on the base. The cone gets slimmer and slimmer so its diameter shortens.

Cylinder given cannot fit inside the cone as the cylinder needs more space.

Thus.

The correct options are:

• C. The volume of the cone is 1/3 the volume of the cylinder.
• E. The cone and cylinder have the same diameter.

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