Two identical cones are inscribed in a cylinder.
Which equation represents the volume of each cone?
A. [tex]$V=\frac{\pi r^2 H}{2}$[/tex]
B. [tex]$V=\frac{\pi r^2 H}{3}$[/tex]
C. [tex]$V=\frac{\pi r^2 H}{4}$[/tex]
D. [tex]$V=\frac{\pi r^2 H}{6}$[/tex]
Answer (2)
The volume of a cone is found using the formula V = 3 1 π r 2 h .
Compare the general formula with the given options.
Identify the option that matches the formula.
The equation representing the volume of each cone is V = 3 π r 2 H .
Explanation
Problem Analysis The problem asks for the formula for the volume of a cone, given that two identical cones are inscribed in a cylinder. We need to identify the correct formula from the given options.
Recall the Volume of a Cone Formula The general formula for the volume of a cone is given by:
V = 3 1 π r 2 h
where:
V is the volume of the cone,
r is the radius of the base of the cone,
h is the height of the cone.
Compare with the given options Comparing the general formula with the given options:
A. V = 2 π r 2 H B. V = 3 π r 2 H C. V = 4 π r 2 H D. V = 6 π r 2 H
Option B, V = 3 π r 2 H , matches the general formula for the volume of a cone, where H is used to represent the height of the cone instead of h .
Final Answer Therefore, the correct equation that represents the volume of each cone is:
V = 3 π r 2 H
Examples
Understanding the volume of a cone is useful in many real-world scenarios. For example, if you're filling a conical paper cup with water, knowing the formula helps you estimate how much liquid it can hold. Similarly, architects and engineers use this formula to calculate the volume of conical structures like roofs or funnels, ensuring they can accurately plan material usage and structural integrity.
The volume of each cone inscribed in the cylinder is represented by the formula V = 3 π r 2 H . This is derived from the standard volume formula for a cone. Therefore, the correct answer is option B.
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