Question

I) what is the ratio of their volume?II) write an expression in terms of r for the volume inside the sphere but outside the cone.

Answer 1

We will have the following:

First, we can see that the volume of the sphere and the regular cone are given by:

`V_s=(4)/(3)\pi r^3`

And:

`V_c=(1)/(3)\pi r^2\ast h`

Now, since the volume of the cone is inscribed by a maximum stablished by the sphere, we know that the maximum height for the cone will be equal to the radius of the sphere, so, we re-write the volume of the cone:

`V_c=(1)/(3)\pi r^2\ast r\Rightarrow V_c=(1)/(3)\pi r^3`

i. Now, we determine the ratio of both volumes as follows:

[Cone to sphere]

`\begin{gathered} r=((1/3)\pi r^3)/((4/3)\pi r^3)\Rightarrow r=((1/3))/((4/3)) \\ \\ \Rightarrow r=(1\ast3)/(4\ast3)\Rightarrow r=(1)/(4) \end{gathered}`

So, the ratio of the volumes of the cone to the sphere will be of 1:4.

ii. And the expression in terms of r for the volume inside the sphere but outside the cone will be:

`V=(4)/(3)\pi r^3-(1)/(3)\pi r^3\Rightarrow V=\pi r^3`

So, the expression is:

`V=\pi r^3`