When a sculptor carves a cone out of a cylinder, where both shapes share the same radius and height, you can describe the situation in terms of their volumes and the relationship between them.
Let's denote the radius and height of both the cone and the cylinder as \( r \) and \( h \), respectively.
Volumes
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Volume of the Cylinder \( V_c \): The volume of a cylinder is given by the formula: \[ V_c = \pi r^2 h \]
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Volume of the Cone \( V_k \): The volume of a cone is given by the formula: \[ V_k = \frac{1}{3} \pi r^2 h \]
Comparison of Volumes
When the cone is carved out of the cylinder, the volume remaining in the cylinder after the cone is removed can be calculated as: \[ V_{\text{remaining}} = V_c - V_k \] Substituting the formulas: \[ V_{\text{remaining}} = \pi r^2 h - \frac{1}{3} \pi r^2 h \] Factoring out \( \pi r^2 h \): \[ V_{\text{remaining}} = \left(1 - \frac{1}{3}\right) \pi r^2 h = \frac{2}{3} \pi r^2 h \]
Summary
- The cylinder has a volume of \( \pi r^2 h \).
- The cone has a volume of \( \frac{1}{3} \pi r^2 h \).
- After carving out the cone, the volume of the remaining material in the cylinder is \( \frac{2}{3} \pi r^2 h \).
This relationship allows the sculptor to understand the proportions of the materials being worked with and can aid in determining how much material will remain after the carving is complete.
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