A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #18 # and the height of the cylinder is #21 #. If the volume of the solid is #20 pi#, what is the area of the base of the cylinder?

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Answer 1 · 2018-05-27T16:55:05

#A_(base) = pi r^2 = 20/27 pi ~~ 2.327 " units"^2#

Given: A solid with a cone on top of a cylinder. #V_(solid) = 20 pi#
#" "h_("cone") = 18; h_("cylinder") = 21; r = r_("cone") = r_("cylinder")#

Equation for the volume of the solid:

#V_(solid) = V_("cylinder") + V_("cone")#

#V_("cylinder") = pi r^2 h_("cylinder"); " "V_("cone") = 1/3 pi r^2 h_("cylinder")#

Substitute into the equation the volume of each solid section:
#V_(solid) = pi r^2 h_("cylinder") + 1/3 pi r^2 h_("cylinder")#

Substitute into the equation the known heights:
#20 pi = 21 pi r^2 + 1/3 * 18 pi r^2#

#20 pi = 21 pi r^2 + 6 pi r^2 = 27 pi r^2#

Divide both sides by #pi: " "20 = 27 r^2#

Solve for #r^2: " "20/27 = r^2#

Calculate the area of the base of the cylinder:

#A_base = pi r^2 = 20/27 pi ~~ 2.327 " units"^2#