To determine the lateral surface area of a cone, which of the following statements or combinations of statements are sufficient?

I. The height of the cone is $12 ext{ cm}$.

II. The radius of the smaller sphere is $\frac{2}{3}$ of the radius of the larger sphere.

III. The radius of the cone is $\frac{5}{6}$ of the radius of the larger sphere, and the difference between the volumes of the two spheres is $\frac{13336}{21} ext{ cm}^3$.

Question

To determine the lateral surface area of a cone, which of the following statements or combinations of statements are sufficient?

I. The height of the cone is $12 	ext{ cm}$.

II. The radius of the smaller sphere is $\frac{2}{3}$ of the radius of the larger sphere.

III. The radius of the cone is $\frac{5}{6}$ of the radius of the larger sphere, and the difference between the volumes of the two spheres is $\frac{13336}{21} 	ext{ cm}^3$.

Accepted Answer

**Solution:**

1. The lateral surface area (LSA) of a cone is given by the formula:
   $$	ext{LSA} = \pi r l$$
   where *r* is the radius of the base and *l* is the slant height.

2. To find *l*, we use the Pythagorean theorem:
   $$l = \sqrt{r^2 + h^2}$$
   where *h* is the height of the cone.

3. Statement I gives the height *h = 12 	ext{ cm}*.

4. Statement II relates the radii of two spheres: radius of smaller sphere = $\frac{2}{3}$ radius of larger sphere.

5. Statement III gives:
   - Radius of cone $r = \frac{5}{6}$ radius of larger sphere.
   - The difference between volumes of the two spheres is $\frac{13336}{21} 	ext{ cm}^3$.

6. Using statements II and III, we can find the radius of the larger sphere by solving the volume difference equation:
   - Volume of a sphere: $V = \frac{4}{3} \pi R^3$
   - Let radius of larger sphere be $R$, smaller sphere radius $= \frac{2}{3} R$
   - Volume difference:
     $$\frac{4}{3} \pi R^3 - \frac{4}{3} \pi \left(\frac{2}{3} Right)^3 = \frac{13336}{21}$$
   - Simplify and solve for $R$.

7. Once $R$ is known, from statement III, radius of cone $r = \frac{5}{6} R$.

8. Using statement I (height) and the radius from above, we can calculate the slant height *l* and then the lateral surface area.

9. Therefore, either the combination of I and III or II and III is sufficient to find the lateral surface area.

10. Statement I and II alone are insufficient because without the radius of the cone or the larger sphere, the lateral surface area cannot be calculated.

**Answer:** Either I and III or II and III are sufficient to answer the question.

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