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The volume of a right cone is
216 pi units
^(3). If its diameter measures 12 units, find its height.
Answer: units

The volume of a right cone is $216 \pi$ units $^{3}$ . If its diameter measures $12$ units, find its height. $\newline$ Answer: units

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Volume of cubes and rectangular prisms: word problems

Full solution

Q. The volume of a right cone is

216 \pi

units

^{3}

. If its diameter measures

12

units, find its height.

\newline

Answer: units

Identify formula for volume: Identify the formula for the volume of a right cone. $\newline$ The volume $V$ of a right cone can be calculated using the formula $V = \frac{1}{3} \pi r^2 h$ , where $r$ is the radius of the base and $h$ is the height of the cone.
Calculate radius from diameter: Given the volume and diameter, calculate the radius of the base of the cone. $\newline$ The diameter of the base is $12$ units, so the radius $(r)$ is half of the diameter, which is $r = \frac{12 \text{ units}}{2} = 6$ units.
Substitute values and solve: Substitute the known values into the volume formula and solve for the height $h$ . We know the volume $V$ is $216 \pi$ cubic units and the radius $r$ is $6$ units. Plugging these values into the formula gives us: $216 \pi = \frac{1}{3} \cdot \pi \cdot (6^2) \cdot h$
Simplify equation for height: Simplify the equation to solve for the height $h$ . First, calculate $6^2$ which is $36$ . Then, the equation becomes: $216 \pi = \left(\frac{1}{3}\right) * \pi * 36 * h$
Remove pi from equation: Remove the $\pi$ from both sides of the equation, as they cancel each other out. $\newline$ This leaves us with: $\newline$ $216 = (\frac{1}{3}) \times 36 \times h$
Multiply to isolate term: Multiply both sides of the equation by $3$ to isolate the term with $h$ . $\newline$ $3 \times 216 = 36 \times h$ $\newline$ $648 = 36 \times h$
Divide to solve for h: Divide both sides of the equation by $36$ to solve for $h$ . $h = \frac{648}{36}$ $h = 18$

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