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:

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

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 18 and the height of the cylinder is 21 21 . If the volume of the solid is 20 pi20 pi, what is the area of the base of the cylinder?