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]$V=\frac{\pi r^2 H}{4}$[/tex]
C. [tex]$V=\frac{\pi r^2 H}{3}$[/tex]
D. [tex]$V=\frac{\pi r^2 H}{6}$[/tex]
Answer (2)
The problem requires finding the correct formula for the volume of a cone.
Recall the general formula for the volume of a cone: V = 3 1 π r 2 h .
Substitute the given height H into the formula: V = 3 1 π r 2 H .
Identify the option that matches the derived formula: V = 3 π r 2 H .
Explanation
Problem Analysis The problem asks for the formula representing the volume of a cone. We know that two identical cones are inscribed in a cylinder. We need to identify the correct formula from the given options.
Recall the formula for the volume of a cone The general formula for the volume V of a cone is given by: V = 3 1 π r 2 h where r is the radius of the base and h is the height of the cone. In this problem, the height of the cone is given as H . Therefore, we can rewrite the formula as: V = 3 1 π r 2 H
Compare with the given options Now, we compare the derived formula with the given options: A. V = 2 π r 2 H B. V = 4 π r 2 H C. V = 3 π r 2 H D. V = 6 π r 2 H
Option C matches the formula we derived.
Final Answer The correct equation representing the volume of each cone is: V = 3 π r 2 H
Examples
Understanding the volume of cones is useful in various real-world applications. For example, when designing ice cream cones, engineers need to calculate the volume accurately to ensure the cone can hold the desired amount of ice cream without overflowing. Similarly, in architecture, the volume of conical roofs or structures needs to be calculated for material estimation and structural stability. Knowing the formula for the volume of a cone, V = 3 1 π r 2 H , helps in these practical scenarios.
The correct equation representing the volume of each cone is V = 3 π r 2 H , as derived from the standard formula for the volume of a cone. This formula matches with option C from the provided choices. Therefore, the answer is option C.
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