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?

Question

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?

Theodore Abad · Accepted Answer

#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")#

Formula for the solid's volume:

#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#

Determine the cylinder's base's area by computing:

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

Sophia Alfred · Answer

undefined The volume of the solid is the sum of the volumes of the cone and the cylinder. Given that the volume of the solid is $20\pi$, the volume of the cone is $\frac{1}{3}\pi r^2 h$ and the volume of the cylinder is $\pi r^2 h$. Substituting the given values for the heights of the cone and the cylinder, and the total volume of the solid, we can solve for the radius $r$. Once we have the radius $r$, we can find the area of the base of the cylinder using the formula for the area of a circle, $A = \pi r^2$.