In the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be [tex]$\frac{\pi}{4}$[/tex] times the volume of the pyramid that it fits inside.
Which expression represents the volume of the cone that is [tex]$\frac{\pi}{4}$[/tex] times the volume of the pyramid that it fits inside?
[tex]$\frac{\pi}{4}\left(2 r^2 h\right)$[/tex]
[tex]$\frac{\pi}{4}\left(4 r^2 h\right)$[/tex]
[tex]$\frac{\pi}{4}\left(\frac{r^2 h}{3}\right)$[/tex]
[tex]$\frac{\pi}{4}\left(\frac{4 r^2 h}{3}\right)$[/tex]
Answer (2)
The problem states the volume of a cone is 4 π times the volume of a pyramid.
Recall the volume of a pyramid with a square base of side 2 r and height h is 3 4 r 2 h .
Multiply the pyramid's volume by 4 π to find the cone's volume: 4 π × 3 4 r 2 h .
The correct expression for the volume of the cone is 4 π ( 3 4 r 2 h ) .
Explanation
Problem Analysis The problem states that the volume of a cone is 4 π times the volume of a pyramid that the cone fits inside. We need to identify the correct expression for the cone's volume, given the options provided.
Expressing the Cone's Volume Let's denote the volume of the pyramid as V p yr ami d . According to the problem, the volume of the cone, V co n e , is given by: V co n e = 4 π V p yr ami d We need to determine which of the given options correctly represents this relationship.
Listing the Options The options are:
4 π ( 2 r 2 h )
4 π ( 4 r 2 h )
4 π ( 3 r 2 h )
4 π ( 3 4 r 2 h )
We need to find which of these expressions is the correct one.
Calculating the Volume Recall that a pyramid with a square base of side length 2 r and height h has a volume of V p yr ami d = 3 1 × ba se × h e i g h t = 3 1 ( 2 r ) 2 h = 3 4 r 2 h Therefore, the volume of the cone is V co n e = 4 π V p yr ami d = 4 π ( 3 4 r 2 h ) This matches option 4.
Final Answer The correct expression for the volume of the cone is 4 π ( 3 4 r 2 h ) .
Examples
Imagine you're designing a party hat in the shape of a cone. Knowing the relationship between a cone's volume and the volume of a related pyramid helps you estimate the amount of material needed. If you know the dimensions of the pyramid that would enclose the cone, you can quickly calculate the cone's volume, ensuring you have enough material to create your party hat. This principle applies to various design and engineering tasks where conical shapes are involved.
The volume of the cone is derived as V co n e = 4 π ( 3 4 r 2 h ) , which simplifies to the correct expression for the cone's volume when compared to the volume of the pyramid it fits inside. Therefore, the correct option is 4 π ( 3 4 r 2 h ) .
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