The volume of a right circular cone, V, in terms of its...

The volume of a right circular cone, $V$, in terms of its radius, $r,$ and its height, $h,$ is given by $V=\frac{1}{3} \pi r^{2} h$ Express $r$ in terms of $h$ if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

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Key Concepts

Unit Consistency and Conversion

This concept emphasizes the importance of ensuring that all measurements in an equation are in compatible units. When volumes and dimensions are given in different units (such as feet for height and inches for volume), careful conversion is necessary to maintain accuracy in calculations.

Volume Formula for a Right Circular Cone

This concept involves understanding that the volume of a right circular cone is given by V = (1/3)?r²h, where r denotes the radius and h denotes the height. It is a key formula in solid geometry used to relate the three dimensions of a cone and is fundamental in problems that require expressing one variable in terms of others.

Algebraic Manipulation of Formulas

This concept pertains to rearranging equations to solve for a specific variable. In the context of the cone volume formula, expressing the radius (r) in terms of the volume and height involves isolating r by performing algebraic operations such as multiplication, division, and taking square roots. This skill is fundamental in both algebra and geometry.

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