Question

a cone and a cylinder have the same base radius and height. if the cylinders volume is 300 cm³, what is the cones volume?
a. 200 cm³
b. 150 cm³
c. 300 cm³
d. 100 cm³

a cone with a radius of 2 cm and a height of 6 cm has a volume of:
a. 8π cm³
b. 4π cm³
c. 12π cm³
d. 10π cm³

Explanation:

Step1: Recall the volume formulas

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h\), and for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). When a cone and a cylinder have the same base radius \(r\) and height \(h\), \(V_{cone}=\frac{1}{3}V_{cylinder}\).
Given \(V_{cylinder} = 300\ cm^{3}\), then \(V_{cone}=\frac{1}{3}\times300\).

Step2: Calculate the cone's volume

\(V_{cone}=\frac{1}{3}\times300 = 100\ cm^{3}\).

Step3: Use the cone - volume formula for the second problem

The volume formula for a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given \(r = 2\ cm\) and \(h=6\ cm\).
Substitute into the formula: \(V=\frac{1}{3}\pi\times(2)^{2}\times6\).
First, calculate \((2)^{2}=4\), then \(\frac{1}{3}\times4\times6\). \(\frac{1}{3}\times24 = 8\). So \(V = 8\pi\ cm^{3}\).

Answer:

For the first problem: d. \(100\ cm^{3}\)
For the second problem: a. \(8\pi\ cm^{3}\)