A sphere is inscribed in a cylinder, as shown below.  Find the ratio of their volumes.


 Jul 7, 2020


Hope that helps.

What is the formula of a sphere and one of a cylinder and assume that the height of both is x. 

 Jul 7, 2020

From the picture we know that the diameter of the sphere is equal to the height and "width" of the cylinder. We know that a sphere's volume is \(\frac{4}{3}\pi r^3\) where \(r\) is the radius. We also know that the cylinder's volume is \(\pi r^2 h\) where \(r\) is the radius and \(h\) is the height. But as we said before (because the sphere is inscribed within the cylinder) that the height and "width" (diameter) is the same so we can rewrite that to \(2\pi r^3\). The ratio of it is \(\frac{\frac{4}{3}\pi r^3}{2\pi r^3}\)\(\pi r^3\) cancels out and we are left with \(\frac{\frac{4}{3}}{2}\) simplify that to \(\Rightarrow\) \(\frac{4}{2\times3} \Rightarrow \frac {4}{6}\) . We can simplify that further to \(\frac{2}{3}\) because both are divisible by 2. So the answer is \(\boxed{\frac{2}{3}}\) or \(\boxed{2:3}\)

 Jul 7, 2020
edited by amazingxin777  Jul 7, 2020

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