Question

if a cone is one - third the volume of a cylinder with the same base and height, what is the formula for its volume?
( \bigcirc ) a. ( v=\frac{1}{3}pi r^{3}h )
( \bigcirc ) b. ( v=\frac{1}{4}pi r^{2}h )
( \bigcirc ) c. ( v=pi rh )
( \bigcirc ) d. ( v=\frac{1}{3}pi r^{2}h )
a cone has a height of 9 cm and a radius of 6 cm. what is its approximate volume using ( piapprox3.14 )?
( \bigcirc ) a. ( 282.6cm^{3} )
( \bigcirc ) b. ( 339.12cm^{3} )
( \bigcirc ) c. ( 452.16cm^{3} )
( \bigcirc ) d. ( 169.56cm^{3} )

Explanation:

Step1: Recall the volume formula of a cylinder

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h\).

Step2: Use the relationship between the cone and cylinder volume

Since the cone is one - third the volume of a cylinder with the same base (\(r\)) and height (\(h\)), then \(V_{cone}=\frac{1}{3}V_{cylinder}\). Substituting \(V_{cylinder}=\pi r^{2}h\) into the formula for the cone's volume, we get \(V_{cone}=\frac{1}{3}\pi r^{2}h\).

Step3: Calculate the volume of the given cone

Given \(r = 6\space cm\), \(h=9\space cm\) and \(\pi = 3.14\). Substitute these values into the formula \(V=\frac{1}{3}\pi r^{2}h\).
First, calculate \(r^{2}\): \(r^{2}=6^{2}=36\).
Then, \(\frac{1}{3}\times3.14\times36\times9\).
\(\frac{1}{3}\times3.14\times36\times9=( \frac{1}{3}\times9)\times3.14\times36\).
\(\frac{1}{3}\times9 = 3\), so \(3\times3.14\times36\).
\(3\times3.14 = 9.42\), then \(9.42\times36=339.12\space cm^{3}\).

Answer:

For the formula of the cone's volume: \(V=\frac{1}{3}\pi r^{2}h\) (corresponds to option a).
For the volume of the cone with \(r = 6\space cm\), \(h = 9\space cm\) and \(\pi=3.14\): \(339.12\space cm^{3}\) (corresponds to option b).