Ask Me Help Desk - Dilated radius means.

This was a test question for one of my sisters and she asked me if I could explain it to her, but I couldn't. Lol. If you guys could, we'd greatly appreciate it!

The radius of a sphere is dilated by a scale factor of 20. Which statement is true?

A. The volume of the new sphere is 200 of the original volume.
B. The volume of the new sphere is 800 of the original volume.
C. The volume of the new sphere is 2000 of the original volume.
D. The volume of the new sphere 8,000 of the original volume.

Ok, my sister chose .B. but she just guessed and she wanted to know if she guessed correctly (lol) or if she was wrong... and why?

Thanks in advance!

Well, the radius of a shere is $4/3 \pi r^3$
So, whatever $r^3$ is is actually $20r^3$ . $20^3$ is 8000, so the answer would be D.

Ok, I'll add something too!!

If you dilate some length, by some number so as to obtain a resultant length the ratio of the initial length to the new length is 1:n, where n is your scale factor.

If you dilate some length, by some number so as to obtain a resultant area the ratio of the initial length to the new area is $1:n^2$ , where n is your scale factor.

If you dilate some length, by some number so as to obtain a resultant volume the ratio of the initial length to the new volume is $1:n^3$ , where n is your scale factor.