To determine the relationship between the dimensions of a cylinder and a cone with the same volume, we can use the volume formulas for both shapes.
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The volume \( V_c \) of a cylinder is given by: \[ V_c = \pi r^2 h_c \] where \( r \) is the radius of the cylinder and \( h_c \) is its height.
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The volume \( V_{cone} \) of a cone is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} \] where \( r \) is the radius of the base of the cone and \( h_{cone} \) is its height.
If both shapes have the same volume, we can set these two volume expressions equal: \[ \pi r^2 h_c = \frac{1}{3} \pi r^2 h_{cone} \]
Dividing both sides by \( \pi r^2 \) (assuming \( r \neq 0 \)): \[ h_c = \frac{1}{3} h_{cone} \]
This means that the height of the cylinder is one-third the height of the cone if both have the same radius.
To find specific values for the radius and height of the cone, you generally need specific values for the cylinder's dimensions or some constraints provided in the context. If you have given options for the dimensions, please provide them, and I can help you determine which might satisfy this proportional relationship.