The image you've provided shows a right circular cone, where:

• The slant height is $8 \, \text{in}$,
• The radius of the base is $4 \, \text{in}$.

To solve any questions regarding this cone (such as finding the volume, surface area, or height), the relevant formulas are:

1. Height of the cone $h$ (which we can find using the Pythagorean theorem).
2. Volume: $V = \frac{1}{3} \pi r^2 h$
3. Surface area (both lateral and total):
   • Lateral surface area: $A_{\text{lateral}} = \pi r l$ where $l$ is the slant height.
   • Total surface area: $A_{\text{total}} = \pi r (r + l)$

Let's first find the height $h$ using the Pythagorean theorem, where $h$, $r$, and $l$ form a right triangle: $h^2 + r^2 = l^2$ $h^2 + 4^2 = 8^2$ $h^2 + 16 = 64$ $h^2 = 64 - 16 = 48$ $h = \sqrt{48} \approx 6.93 \, \text{in}$

Now, you can use this height to calculate other properties like the volume or surface area. Would you like to calculate the volume, surface area, or explore more details?

Follow-up Questions:

1. What is the total surface area of this cone?
2. What is the volume of the cone using the height calculated?
3. How does changing the radius affect the volume of the cone?
4. What would the lateral surface area be if the slant height were doubled?
5. How would the height change if the slant height was increased to 10 inches?

Tip:

When solving for geometric properties of a cone, always ensure to use the correct height for volume calculations, not the slant height.