Answer :
Let's walk through the steps to understand how to find the ratio of the volumes of two similar cones with given radii.
1. Understand the relationship between similar cones:
- The volumes of similar cones are proportional to the cubes of their corresponding radii. This is because the volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
If two cones are similar, their heights are also proportional to their radii. Hence the ratio of their volumes is:
[tex]\[ \left( \frac{r_1}{r_2} \right)^3 \][/tex]
2. Given values:
- The radius of the larger cone [tex]\( r_1 \)[/tex] is 6.
- The radius of the smaller cone [tex]\( r_2 \)[/tex] is 1.
3. Calculate the volumes' ratio:
- Using the relationship mentioned above, the ratio of the volumes of the cones is:
[tex]\[ \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{6}{1} \right)^3 = 6^3 = 216 \][/tex]
4. Conclusion:
- Therefore, the ratio of the volumes of the larger cone to the smaller cone is [tex]\( 216:1 \)[/tex].
So the correct answer is D. 216:1.
1. Understand the relationship between similar cones:
- The volumes of similar cones are proportional to the cubes of their corresponding radii. This is because the volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
If two cones are similar, their heights are also proportional to their radii. Hence the ratio of their volumes is:
[tex]\[ \left( \frac{r_1}{r_2} \right)^3 \][/tex]
2. Given values:
- The radius of the larger cone [tex]\( r_1 \)[/tex] is 6.
- The radius of the smaller cone [tex]\( r_2 \)[/tex] is 1.
3. Calculate the volumes' ratio:
- Using the relationship mentioned above, the ratio of the volumes of the cones is:
[tex]\[ \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{6}{1} \right)^3 = 6^3 = 216 \][/tex]
4. Conclusion:
- Therefore, the ratio of the volumes of the larger cone to the smaller cone is [tex]\( 216:1 \)[/tex].
So the correct answer is D. 216:1.
