Question

problem 7

here is another cylinder and cone with the same height and the same-sized bases.

determine the volume of the cone.

\\(v = 30\pi\\)

\\(v = ?\\)

Explanation:

Identify the given values and formulas

We are given a cylinder with a volume of \(V_{\text{cylinder}} = 30\pi\).
We are also given a cone that has the same height \(h\) and the same-sized base (radius \(r\)) as the cylinder.
The formula for the volume of a cylinder is:

$$V_{\text{cylinder}} = \pi r^2 h$$

The formula for the volume of a cone is:

$$V_{\text{cone}} = \frac{1}{3}\pi r^2 h$$

Relate the two volumes

Since the cylinder and the cone share the same height \(h\) and the same base radius \(r\), we can substitute the cylinder volume formula into the cone volume formula:

$$V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}$$

Calculate the volume of the cone

Substitute the given volume of the cylinder \(V_{\text{cylinder}} = 30\pi\) into the relation:

$$V_{\text{cone}} = \frac{1}{3} \times 30\pi = 10\pi$$

Answer:

Determine the volume of the cone.
<blank>\(10\pi\)</blank>