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The volume of a right cone is
36 pi units
^(3). If its circumference measures
6pi units, find its height.
Answer: units

The volume of a right cone is $36 \pi$ units $^{3}$ . If its circumference measures $6 \pi$ units, find its height. $\newline$ Answer: units

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Volume of cylinders

Full solution

Q. The volume of a right cone is

36 \pi

units

^{3}

. If its circumference measures

6 \pi

units, find its height.

\newline

Answer: units

Identify Formula and Values: Identify the formula for the volume of a cone and the given values. $\newline$ The volume of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$ , where $V$ is the volume, $r$ is the radius, and $h$ is the height. We are given the volume $V = 36 \pi$ units $^3$ .
Identify Circumference Formula: Identify the formula for the circumference of a circle and the given value. $\newline$ The circumference of a circle is given by the formula $C = 2 \cdot \pi \cdot r$ , where $C$ is the circumference and $r$ is the radius. We are given the circumference $C = 6 \pi$ units.
Calculate Radius from Circumference: Calculate the radius of the base of the cone using the circumference. $\newline$ Using the formula $C = 2 \pi r$ , we can solve for $r$ : $\newline$ $6 \pi = 2 \pi r$ $\newline$ Divide both sides by $2 \pi$ : $\newline$ $r = \frac{6 \pi}{2 \pi}$ $\newline$ $r = 3$ units
Substitute Radius into Volume Formula: Substitute the radius into the volume formula and solve for the height. $\newline$ We have $V = \frac{1}{3} \pi r^2 h$ and $r = 3$ units. Substitute these values into the volume formula: $\newline$ $36 \pi = \frac{1}{3} \pi (3)^2 h$
Simplify Equation and Solve: Simplify the equation and solve for $h$ . $36 \pi = (\frac{1}{3}) \times \pi \times 9 \times h$ Multiply both sides by $3$ to get rid of the fraction: $108 \pi = \pi \times 9 \times h$ Divide both sides by $\pi \times 9$ : $h = (\frac{108 \pi}{\pi \times 9})$ $h = 12$ units

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