Question #bb3d8

1 Answer
Jul 5, 2016

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