Two identical cones are inscribed in a cylinder.
Which equation represents the volume of each cone?
A. [tex]$V=\frac{\pi r^2 H}{4}$[/tex]
B. [tex]$V=\frac{\pi r^2 H}{2}$[/tex]
C. [tex]$V=\frac{\pi r^2 H}{6}$[/tex]
D. [tex]$V=\frac{\pi r^2 H}{3}$[/tex]
Answer (1)
The volume of a cone is given by V = 3 1 π r 2 h .
Compare the given options with the formula V = 3 1 π r 2 H .
Option D, V = 3 π r 2 H , matches the formula.
The correct equation representing the volume of each cone is V = 3 π r 2 H .
Explanation
Problem Analysis The problem asks for the volume of a cone inscribed in a cylinder. We need to identify the correct formula for the volume of a cone 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 of the cone and h is the height of the cone. In this problem, the height of the cone is given as H . So, we can rewrite the formula as: V = 3 1 π r 2 H
Compare the options with the formula Now, let's compare the given options with the correct formula:
A. V = 4 π r 2 H - Incorrect, the coefficient should be 3 1 , not 4 1 .
B. V = 2 π r 2 H - Incorrect, the coefficient should be 3 1 , not 2 1 .
C. V = 6 π r 2 H - Incorrect, the coefficient should be 3 1 , not 6 1 .
D. V = 3 π r 2 H - Correct, this is equivalent to V = 3 1 π r 2 H .
Examples
Understanding the volume of cones is essential 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 conical roofs of towers or tents require precise volume calculations to determine the amount of material needed for construction. Knowing the formula for the volume of a cone, V = 3 1 π r 2 H , allows for efficient and cost-effective designs in these scenarios.
