Surface Volume Formulas
1. The problem asks to find formulas for points 8, 9, 10 related to cones, spheres, and hemispheres.
2. Based on Chapter 11.5 summary, points 1-7 are about surface areas and volumes.
3. We can reasonably assume points 8, 9, 10 involve important related formulas not listed, commonly:
- 8. Slant height relation in cone: $l = \sqrt{r^2 + h^2}$, where $r$ is the base radius, $h$ is height, and $l$ is slant height.
- 9. Volume of hemisphere: Half the volume of a sphere, so
$$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$
- 10. Surface area of right circular cone including base: Total area $= \pi r (l + r)$ (already point 2). Another useful formula could be the lateral surface area of frustum, or you may want to consider:
Curved surface area of frustum = $\pi (r_1 + r_2) l$, where $r_1$ and $r_2$ are radii of the two circular ends, $l$ is slant height.
4. Explanation:
- For 8, slant height $l$ relates the radius and height by the Pythagorean theorem.
- For 9, volume of hemisphere is half the sphere's volume.
- For 10, the curved surface area of a conical frustum formula is another important extension.
Final answers:
8. $l = \sqrt{r^2 + h^2}$
9. $V = \frac{2}{3} \pi r^3$
10. Curved Surface Area of frustum $= \pi (r_1 + r_2) l$
These complete the summary with important related formulas.