To analyze the relationship between the volumes of the sphere, cylinder, and cone, we need to use the volume formulas for each shape.
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Volume of a Sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of a Cylinder (V_c): \[ V_c = \pi r^2 h \] Given that the height \( h \) is twice the radius \( r \) (i.e., \( h = 2r \)): \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of a Cone (V_cone): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Again, with \( h = 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's evaluate the relationships:
- Comparing the volumes:
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The volume of the cylinder: \[ V_c = 2\pi r^3 \]
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The volume of the cone: \[ V_{\text{cone}} = \frac{2}{3}\pi r^3 \]
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Calculating the sum of the volumes of the cone: \[ 3 \cdot V_{\text{cone}} = 3 \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c \]
This shows that: \[ 3 \text{ cones} = 1 \text{ cylinder} \]
Now let's compare the sphere's volume:
- The sphere's volume is: \[ V_s = \frac{4}{3} \pi r^3 \]
The primary relationships we derived are:
- \( 3 \text{ cones} = 1 \text{ cylinder} \)
- \( \frac{4}{3} \pi r^3 \) (volume of the sphere) is different from the sum of the cylinder and cone.
Accordingly, the true relationship among the options provided is: 3 cones = 1 cylinder.
So, the correct answer is: 3 cones = 1 cylinder.
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