Question

A cone has a diameter of `18\ `units and a slant height of `15\ `units.

Answer 1

To find various properties of the cone specified, let's first gather the information we have:

- Diameter of the cone, \(d = 18\) units
- Radius of the cone, \(r = \frac{d}{2} = \frac{18}{2} = 9\) units
- Slant height of the cone, \(l = 15\) units

Using this information, we can find various properties such as the height and volume of the cone.

### 1. Finding the Height of the Cone

We can use the Pythagorean theorem to find the height (\(h\)) of the cone. The relation between the radius, height, and slant height in a right triangle formed by the radius, height, and slant height is given by:

\[
l^2 = r^2 + h^2
\]

Substituting the known values:
\[
15^2 = 9^2 + h^2
\]
\[
225 = 81 + h^2
\]
\[
h^2 = 225 - 81
\]
\[
h^2 = 144
\]
\[
h = \sqrt{144} = 12 \text{ units}
\]

### 2. Finding the Volume of the Cone

The volume (\(V\)) of a cone can be calculated using the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

Now substituting \(r = 9\) and \(h = 12\):

\[
V = \frac{1}{3} \pi (9^2)(12)
\]
\[
V = \frac{1}{3} \pi (81)(12)
\]
\[
V = \frac{1}{3} \pi (972)
\]
\[
V = 324 \pi \text{ cubic units}
\]

### 3. Finding the Surface Area of the Cone

The surface area (\(A\)) of a cone is given by the formula:

\[
A = \pi r l + \pi r^2
\]

The first term, \(\pi r l\), represents the lateral surface area, and the second term, \(\pi r^2\), represents the area of the base.

Substituting \(r = 9\) and \(l = 15\):

\[
A = \pi(9)(15) + \pi(9^2)
\]
\[
A = 135\pi + 81\pi
\]
\[
A = 216\pi \text{ square units}
\]

### Summary

- Radius \(r = 9\) units
- Height \(h = 12\) units
- Volume \(V = 324\pi \approx 1017.88\) cubic units (using \(\pi \approx 3.14\))
- Surface Area \(A = 216\pi \approx 678.24\) square units (using \(\pi \approx 3.14\))