To determine the correct relationship between the volumes of the sphere, cylinder, and cone, let's first calculate the volumes of each shape using the given parameters.
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Definitions:
- Let the radius \( r \) be \( r \).
- The height \( h \) of the cylinder and cone is \( 2r \).
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Volume Formulas:
- Volume of a sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- Volume of a cylinder: \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \]
- Volume of a cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
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Comparison of Volumes:
- Now, let's evaluate the relationships:
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Sphere vs Cylinder and Cone: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] \[ V_{\text{cylinder}} + V_{\text{cone}} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] Here, \( V_{\text{sphere}} \neq V_{\text{cylinder}} + V_{\text{cone}} \).
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Comparing 3 cones and 1 cylinder: \[ 3 \times V_{\text{cone}} = 3 \times \frac{2}{3} \pi r^3 = 2 \pi r^3 = V_{\text{cylinder}} \] Thus, \( 3 \text{ cones} = 1 \text{ cylinder} \).
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Comparing 1 cylinder with 2 cones and 1 sphere: \[ 1 \text{ cylinder} = 2 \text{ cones} + 1 \text{ sphere} \] Let's check: \[ 2 \times \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] Thus, the equality does not hold.
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Comparing sphere and cone to cylinder: \[ V_{\text{sphere}} - V_{\text{cone}} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \text{ which does not equal } V_{\text{cylinder}} (= 2 \pi r^3). \]
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- Now, let's evaluate the relationships:
The only true relationship we have derived is:
3 cones = 1 cylinder
Thus, the response that is true about the relationship is:
- 3 cones = 1 cylinder.