To determine the relationship between the volumes of a sphere, cylinder, and cone with the given conditions, we can use the formulas for the volumes of these shapes:
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Sphere: The volume \( V \) is given by the formula: \[ V = \frac{4}{3} \pi r^3 \]
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Cylinder: The volume \( V \) is given by the formula: \[ V = \pi r^2 h \] Here, since the height \( h \) of the cylinder is twice the radius \( r \), we can substitute \( h \): \[ V = \pi r^2 (2r) = 2 \pi r^3 \]
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Cone: The volume \( V \) is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] Again, substituting \( h = 2r \): \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we have:
- Sphere Volume: \( \frac{4}{3} \pi r^3 \)
- Cylinder Volume: \( 2 \pi r^3 \)
- Cone Volume: \( \frac{2}{3} \pi r^3 \)
Now let's compare these volumes to find the relationships:
Finding the relationships:
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Comparing Cylinder to Cone: \[ \text{Cylinder volume} = 2 \pi r^3 \] \[ \text{Cone volume} = \frac{2}{3} \pi r^3 \] So, \[ 2 \text{ cones} = 2 \cdot \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 = \text{Sphere volume} \] Thus, \( 2 \text{ cones} = 1 \text{ sphere} \).
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Comparing 2 Cones to Cylinder: \[ 2 \text{ cones} = 2 \cdot \left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 = \text{ Sphere volume}, \text{ so this does not hold.} \]
Now substituting the findings into the options given:
- 2 cones = 1 cylinder: False
- 2 spheres = 1 cone: False
- 2 cones = 1 sphere: True
- 2 cylinders = 1 cone: False
Therefore, the correct answer is: 2 cones = 1 sphere.
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