Question

the volume of the cone shown in the figure below is \\( \frac{1}{3} \\) the volume of the cylinder. in terms of \\( h \\), what is the volume of the cone? figure: cylinder with diameter 8, height \\( h \\), and a cone (dashed lines) inside. multiple - choice options (partially visible) below the figure.

Explanation:

Step1: Find the radius of the cylinder (and cone, since they share the same base)

The diameter of the cylinder is 8, so the radius \( r = \frac{8}{2} = 4 \).

Step2: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = \pi r^2 h \). Substituting \( r = 4 \), we get \( V_{\text{cylinder}} = \pi (4)^2 h = 16\pi h \).

Step3: Calculate the volume of the cone

The volume of the cone is \( \frac{1}{3} \) of the volume of the cylinder. So \( V_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}} = \frac{1}{3} \times 16\pi h = \frac{16}{3}\pi h \).

Answer:

\(\frac{16}{3}\pi h\)