Answer:
71π cubic inches
Step-by-step explanation:
Given a cylinder with radius r and height h the volume = π · r² · h
The base area = π · r²
For a cone with the same radius and hence the same base area and a height of h, Volume = 1/3 x π · r² · h
So volume of cone = 1/3 x volume of equivalent cylinder
Given volume of cylinder = 213π cubic inches, the volume of cone with same base area
= 1/3 x 213π cubic inches
= 71π cubic inches
Answer:
71π cubic inches
Step-by-step explanation:
The formulas for the volume of a cone and the volume of a cylinder are:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cone}}\\\\V=\dfrac{1}{3}\pi r^2h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cylinder}}\\\\V=\pi r^2 h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
Given that the cone and cylinder share the same base area and height, the values of r and h will be the same for both formulas. Therefore, the volume of the cone with the same base area and height as the cylinder is simply one-third of the volume of the cylinder:
[tex]V_{\text{cone}} = \dfrac{1}{3} V_{\text{cylinder}}[/tex]
Substituting the given volume of the cylinder (213π cubic inches) into the equation, we get:
[tex]V_{\text{cone}} = \dfrac{1}{3} \cdot 213\pi \\\\\\ V_{\text{cone}} = 71\pi\; \rm cubic\;inches[/tex]
So, the volume of the cone with the same base area and height as the cylinder is 71π cubic inches.
