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The volume of a right cone is
700 pi units
^(3). If its height is 21 units, find its diameter.

The volume of a right cone is $700\pi$ units $^3$ . If its height is $21$ units, find its diameter.

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Multi-step problems with unit conversions

Full solution

Q. The volume of a right cone is

700\pi

units

^3

. If its height is

21

units, find its diameter.

Volume Formula Application: The formula for the volume of a right cone is $V = \frac{1}{3} \pi r^2 h$ , where $V$ is the volume, $r$ is the radius, and $h$ is the height. We are given the volume ( $V = 700 \pi$ ) and the height ( $h = 21$ units). We need to solve for the radius $r$ first.
Substitute and Solve for $r^2$ : Substitute the given values into the volume formula and solve for $r^2$ . $\newline$ $700 \pi = (\frac{1}{3}) \cdot \pi \cdot r^2 \cdot 21$ $\newline$ To isolate $r^2$ , we first divide both sides by $\pi$ , which cancels out the $\pi$ terms. $\newline$ $700 = (\frac{1}{3}) \cdot r^2 \cdot 21$
Isolate $r^2$ : Next, we divide both sides by $21$ to further isolate $r^2$ . $\newline$ $\frac{700}{21} = \left(\frac{1}{3}\right) * r^2$ $\newline$ $r^2 = \left(\frac{700}{21}\right) * 3$ $\newline$ $r^2 = 100$
Solve for r: Now, we take the square root of both sides to solve for r. $\newline$ $r = \sqrt{100}$ $\newline$ $r = 10$
Find Diameter: Since the diameter is twice the radius, we multiply the radius by $2$ to find the diameter. $\newline$ Diameter = $2 \times r$ $\newline$ Diameter = $2 \times 10$ $\newline$ Diameter = $20$ units

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