To analyze the relationship between the volumes of the sphere, cylinder, and cone, let’s first find the volume formulas for each of the shapes given their shared dimensions.
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Sphere The volume \( V_s \) of a sphere with radius \( r \) is given by: \[ V_s = \frac{4}{3} \pi r^3 \]
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Cylinder The volume \( V_c \) of a cylinder with radius \( r \) and height \( h = 2r \) is given by: \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Cone The volume \( V_{cone} \) of a cone with radius \( r \) and height \( h = 2r \) is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we have the volume formulas:
- Volume of the sphere \( V_s = \frac{4}{3} \pi r^3 \)
- Volume of the cylinder \( V_c = 2 \pi r^3 \)
- Volume of the cone \( V_{cone} = \frac{2}{3} \pi r^3 \)
Now we can compare the volumes to check the given relationships:
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Checking cone = cylinder - sphere: \[ \frac{2}{3} \pi r^3 \stackrel{?}{=} 2 \pi r^3 - \frac{4}{3} \pi r^3 \] Simplifying the right side: \[ 2 \pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This is true.
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Checking cone = sphere - cylinder: \[ \frac{2}{3} \pi r^3 \stackrel{?}{=} \frac{4}{3} \pi r^3 - 2 \pi r^3 \] Simplifying the right side: \[ \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3 \] This is false.
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Checking cylinder = sphere - cone: \[ 2 \pi r^3 \stackrel{?}{=} \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 \] Simplifying the right side: \[ \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This is false.
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Checking sphere = cylinder + cone: \[ \frac{4}{3} \pi r^3 \stackrel{?}{=} 2 \pi r^3 + \frac{2}{3} \pi r^3 \] Simplifying the right side: \[ 2 \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This is false.
From these calculations, we conclude that the only true relationship is:
1. cone = cylinder – sphere