To find the ratio of the volumes of two similar cones, we can use the fact that the volumes of similar shapes are proportional to the cube of the ratio of their corresponding linear dimensions (such as radii or heights).
Given that the radii of the two similar cones are 6 and 1, respectively, let's calculate the ratio of their volumes step-by-step:
1. Determine the ratio of the radii:
- The radius of the first cone, [tex]\(r_1\)[/tex], is 6.
- The radius of the second cone, [tex]\(r_2\)[/tex], is 1.
- The ratio of the radii is [tex]\( \frac{r_1}{r_2} = \frac{6}{1} = 6 \)[/tex].
2. Use the volume ratio formula:
- The volumes of similar cones are proportional to the cubes of the ratios of their corresponding linear dimensions.
- Therefore, the ratio of the volumes [tex]\(V_1\)[/tex] to [tex]\(V_2\)[/tex] is given by [tex]\( \left( \frac{r_1}{r_2} \right)^3 \)[/tex].
3. Calculate the ratio of the volumes:
- Substitute in the ratio of the radii:
[tex]\[
\left( \frac{r_1}{r_2} \right)^3 = \left( \frac{6}{1} \right)^3 = 6^3 = 216
\][/tex]
4. Express the ratio:
- The ratio of the volumes of the two cones is 216:1.
Hence, the ratio of their volumes is [tex]\(216:1\)[/tex]. Therefore, the correct answer is:
C. 216:1
