A cone and sphere have equal volumes and radii of equal length. If the height of the cone is 36 centimeters, then what is the length of the radius of each shape in centimeters?

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Write Formulas: The volume of a cone is given by the formula $ V_{cone} = \frac{1}{3} \pi r^2 h $, where $ r $ is the radius and $ h $ is the height. The volume of a sphere is given by the formula $ V_{sphere} = \frac{4}{3} \pi r^3 $. Since the volumes are equal, we can set the two formulas equal to each other and solve for $ r $.Set Equal: Set the volume of the cone equal to the volume of the sphere: $\frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3$.Substitute Height: Substitute the given height $ h = 36 $ cm into the equation: $\frac{1}{3} \pi r^2 \times 36 = \frac{4}{3} \pi r^3$.Simplify Equation: Simplify the equation by multiplying out the constants: $12 \pi r^2 = 4 \pi r^3$.Isolate Radius: Divide both sides of the equation by $ 4 \pi $ to isolate $ r $ on one side: $3 r^2 = r^3$.Solve for Radius: Divide both sides of the equation by $ r^2 $, assuming $ r \neq 0 $: $3 = r$.Final Result: We find that the radius $ r $ is 3 centimeters.

A cone and sphere have equal volumes and radii of equal length. If the height of the cone is $36$ centimeters, then what is the length of the radius of each shape in centimeters?

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A cone and sphere have equal volumes and radii of equal length. If the height of the cone is $36$ centimeters, then what is the length of the radius of each shape in centimeters?