Question

Question #bb3d8

Answer 1

`V = 7/9pi r^2 H`

Explanation:

Volume of this solid is a sum of two volumes - the one of a cone and that of a cylinder.

Since we know the radius of both, all we need is the height of each - `h_1` (height of a cone) and `h_2` (height of a cylinder).
We do not know these heights but we know two important equations they participate in:
(1) `h_2 = 2h_1`
(2) `h_1 + h_2 = H`
where `H` is a known height of an entire solid.

From the two equations above we can easily find `h_1` and `h_2` in terms of `H`:
substituting (1) into (2), we get
`h_1+2h_1 = H`
`rArr` `h_1 = H/3`
`rArr` `h_2 = (2H)/3`

Knowing heights and radiuses of a cone and a cylinder, we can calculate each volume.
The volume of a cone is
`V_1 = 1/3 pi r^2h_1 = 1/3 pi r^2 H/3 = (pi r^2H)/9`
The volume of a cylinder is
`V_2=pir^2h_2 = pi r^2 (2H)/3`

Total volume of a solid is
`V=V_1+V_2 =`
`= pir^2H(1/9+2/3) =`
`= 7/9pi r^2 H`