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#