Question

the volume of a right cone is 343π units³. if its height is 21 units, find its diameter.

answer

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Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a right cone is given by 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.
We know that \( V = 343\pi \) and \( h=21 \). Substitute these values into the formula:
\( 343\pi=\frac{1}{3}\pi r^{2}(21) \)

Step2: Simplify the equation to solve for \( r^{2} \)

First, we can cancel out \( \pi \) from both sides of the equation:
\( 343=\frac{1}{3}\times21\times r^{2} \)
Simplify \( \frac{1}{3}\times21 \), which is equal to \( 7 \). So the equation becomes:
\( 343 = 7r^{2} \)
Then, divide both sides by \( 7 \):
\( r^{2}=\frac{343}{7}=49 \)

Step3: Solve for the radius \( r \)

Take the square root of both sides. Since \( r \) represents the radius (a non - negative quantity), we have:
\( r=\sqrt{49} = 7 \)

Step4: Find the diameter \( d \)

The diameter \( d \) of a circle is related to the radius \( r \) by the formula \( d = 2r \). Substitute \( r = 7 \) into this formula:
\( d=2\times7 = 14 \)

Answer:

14