Question

A sphere fits exactly inside a cylinder. What is the relationship between their volumes? Surface areas? Image attached.

Answer 1

Proofs get awarded bonus points (which aren't worth anything).

Answer 2

The radius of the ball and the radius of the cylinder have to be equal.
Also the height of the cylinder has to be the diameter of the ball, so 2r.
Their volumes will be:
\[V_{ball} = \frac{4}{3}\pi r^3 \]\[V_{cylinder} = \pi r^2h = \pi r^2 (2r) = 2\pi r^3\]
If you want a ratio for an answer you would divide them:
\[\frac{V_{ball}}{V_{cylinder}} = \frac{\frac{4}{3}\pi r^3}{2\pi r^3} = \frac{4}{3}*\frac{1}{2} = \frac{2}{3}\]
So the ball takes up 2/3 of the space in the cylinder.

Answer 3

Looking at Surface Area we get:
\[SA_{ball} = 4\pi r^2\]\[SA_{cylinder} = 2\pi r^2+2\pi rh = 2\pi r^2+2\pi r(2r) = 2\pi r^2+4\pi r^2 = 6\pi r^2\]
Their ratio will be:
\[\frac{SA_{ball}}{SA_{cylinder}} = \frac{4\pi r^2}{6\pi r^2} = \frac{2}{3}\]

Answer 4

Nice job, very thorough!