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# help

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

Jul 7, 2020

#1
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2:3

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
#2
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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