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 #36 # and the height of the cylinder is #6 #. If the volume of the solid is #48 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 #36  # and the height of the cylinder is #6   #.  If the volume of the solid is #48 pi#, what is the area of the base of the cylinder?

Sophia Alfred · Accepted Answer

#A=2.667pi#

#A = 8.37#

We can use the volume of the shape to solve for the radius using the combination of the volume of the cylinder and the cone.

Volume of a Cone = #1/3pir^2h#
Volume of a Cylinder = #pir^2h#

Total Volume = #1/3pir^2h+pir^2h#

#48pi = 1/3pir^2(36)+pir^2(6)#

#48pi = 12pir^2+6pir^2#

#48pi = 18pir^2#

#(48cancelpi)/(18cancelpi) = ((cancel18cancelpi)r^2)/(cancel18cancelpi)#

#2.667 = r^2#

#sqrt2.667 = sqrt(r^2)#

#1.633 = r#

Area of the circular base is #A=pir^2#

#A=pi(1.633)^2#

#A=2.667pi#

#A = 8.37#

Allison Allen · Answer

undefined Since the volume of the solid is given as 48π and it consists of a cone on top of a cylinder, the volume can be represented as the sum of the volumes of the cone and the cylinder. The volume of a cone is $\frac{1}{3}πr^2h$ and the volume of a cylinder is $πr^2h$. Given the height of the cone (36) and the height of the cylinder (6), we can write the equation:

$$\frac{1}{3}πr^2(36) + πr^2(6) = 48π$$

Solving this equation, we find the radius ($r$) of the cone. Once we have $r$, we can calculate the area of the base of the cylinder, which is $πr^2$.