If two solids have equal cross-sectional areas at every level parallel to the respective bases, then the two solids have equal volume.
The two shared solids both have a height of [tex]$2 r$[/tex] units. At every level, the areas of the cross sections of both solids equal [tex]$\pi(r^2-b^2)$[/tex].
Cross-section area [tex]$=\pi r^2-\pi b^2 =\pi(r^2-b^2)$[/tex]
Which of the following can be derived by writing an expression that represents the volume of:
A. one cone within the cylinder.
B. the two cones within the cylinder.
C. the solid between the two cones and the cylinder.
D. the cylinder.
Answer (2)
The problem involves understanding Cavalieri's principle, which states that solids with equal height and cross-sectional areas have equal volumes.
The cross-sectional area is given as π ( r 2 − b 2 ) , suggesting a cylinder with a portion removed.
The volume can be derived by considering the entire cylinder and accounting for the subtracted area at each level.
The expression that represents the volume is the cylinder. the cylinder
Explanation
Problem Analysis and Cavalieri's Principle The problem states that two solids have equal height ( 2 r ) and equal cross-sectional areas π ( r 2 − b 2 ) at every level. According to Cavalieri's principle, if two solids have the same height and the same cross-sectional area at every level, then they have the same volume. The cross-sectional area π ( r 2 − b 2 ) suggests that we are dealing with a cylinder of radius r from which a volume corresponding to a circle of radius b has been removed. The question asks us to identify which expression represents the volume of the solid.
Volume of the Cylinder Let's consider a cylinder with radius r and height 2 r . The volume of this cylinder is given by: V cy l in d er = π r 2 h = π r 2 ( 2 r ) = 2 π r 3
Volume of Two Cones Now, let's consider the possibility of two cones within the cylinder. If we have two cones, each with radius r and height r , their combined volume would be: V 2 co n es = 2 × 3 1 π r 2 h = 2 × 3 1 π r 2 ( r ) = 3 2 π r 3
Volume of the Solid Between the Cones and Cylinder The solid between the two cones and the cylinder would have a volume equal to the volume of the cylinder minus the volume of the two cones: V so l i d = V cy l in d er − V 2 co n es = 2 π r 3 − 3 2 π r 3 = 3 4 π r 3 However, the cross-sectional area given is π ( r 2 − b 2 ) , which suggests that at each level, we are subtracting an area of π b 2 from the cylinder's cross-sectional area π r 2 . This could represent the area of a hole or a removed section within the cylinder.
Deriving the Volume Expression Since the cross-sectional areas of the two solids are equal at every level, and one of the solids is a cylinder, the other solid must also have the same volume as a cylinder with adjusted cross-sections. The expression that represents the volume is that of the cylinder, but with a modification to account for the subtracted area at each level. However, the question is asking which of the given options represents the volume. Given the context of Cavalieri's principle and the equal cross-sectional areas, the volume can be derived by considering the entire cylinder and then subtracting the appropriate volumes to account for the 'holes' represented by the b 2 term in the cross-sectional area.
Final Answer Based on the problem statement and the application of Cavalieri's principle, the expression that represents the volume is best described by considering the entire cylinder. The cross-sectional area π ( r 2 − b 2 ) indicates that at each level, we have the area of the cylinder's cross-section ( π r 2 ) minus some area ( π b 2 ). Therefore, the volume is derived by considering the entire cylinder.
Conclusion The expression that represents the volume is the cylinder.
Examples
Cavalieri's principle is useful in architecture and engineering when calculating the volume of complex shapes. For example, when designing a building with irregular cross-sections, architects can use Cavalieri's principle to ensure that different designs have the same volume, even if their shapes vary. This is particularly useful when dealing with structures that have varying widths or heights, ensuring that the overall volume and material usage remain consistent across different design iterations. This principle helps in optimizing material usage and structural integrity while allowing for creative design variations.
The expression that represents the derived volume based on Cavalieri's principle is that of the cylinder, given its height and cross-sectional area at every level. This expression aligns with the inner workings of volumes derived from solids with equal cross-sections. Therefore, the option selected is D. the cylinder.
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