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 #9 # and the height of the cylinder is #6 #. If the volume of the solid is #135 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 #9  # and the height of the cylinder is #6   #.  If the volume of the solid is #135 pi#, what is the area of the base of the cylinder?

Noah Atkinson · Accepted Answer

#15pi#

Cone volume #(1/3)pir^2(h_1) = (1/3)pir^2(9)=3pir^2#

Cylinder volume #=pir^2(h_2)=pir^2xx6=6pir^2#

Cone volume plus cylinder volume equals solid volume.

#=> 135pi = 3pir^2+6pir^2=9pir^2#

#=> r^2=135/9 =15#

cylinder base area #= pir^2 = 15pi#

Option 2)

The base area of the cylinder and the cone is equal because they have the same radius.

Let #A_(base)# be the base area

Cone volume #1/3A_(base)h_1=1/3A_(base)xx9=3A_(base)#

Cylinder volume #A_(base)h_2=6A_(base)#

Cone volume plus cylinder volume equals solid volume.

#=> 135pi=3A_(base)+6A_(base)=9A_(base)# #=> A_(base) = (135pi)/9=15pi#

Violet Aldrich · Answer

undefined First, we find the volume of the solid:

Volume of cone = $ \frac{1}{3} \pi r^2 h $
Volume of cylinder = $ \pi r^2 h $

Given:
Height of cone, $ h_{cone} = 9 $
Height of cylinder, $ h_{cylinder} = 6 $
Total volume = $ 135\pi $

From the given data, we can write:
$ \frac{1}{3} \pi r^2 \times 9 + \pi r^2 \times 6 = 135\pi $

Solve this equation to find the radius ($ r $). Then, once you have the radius, you can find the area of the base of the cylinder using the formula $ A = \pi r^2 $.