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 #66 # and the height of the cylinder is #5 #. If the volume of the solid is #64 pi#, what is the area of the base of the cylinder?

2 Answers
Jun 6, 2018

#A=64/27pi# #u^2#

Explanation:

The volume of the cone is given by: #v=1/3pir^2h#
Since the height of the cone is 66, then #h=66#
So, #v=1/3pir^2times66=22pir^2#

The volume of a cylinder is given by: #v=pir^2h#
Since the height of the cylinder is 5, then #h=5#
So, #v=pir^2h = pir^2times5=5pir^2#

The total volume of the solid is #64pi#
Therefore, #22pir^2+5pir^2=64pi#

#27pir^2=64pi#
#r^2=(64pi)/(27pi)#
#r^2=64/27#
#r=+-8/(3sqrt3)#
Since r is the radius, it must be have the restriction: #r>0#
Therefore, #r=8/(3sqrt3)#units

To find the base of the cylinder, we need to know that the base is a circle. The area of a circle is given by #A=pir^2 = pitimes(8/(3sqrt3))^2=64/27pi# #u^2#

Jun 6, 2018

The area of the base of the cylinder is: #A=pir^2=(64pi)/27#

Explanation:

The area of the base we need to find is: #A=pir^2#, where #r# is the radius of the cylinder.

The volume of the cylinder is: #pir^2*h_1#
where #h_1# is the height of the cylinder.
The volume of the cone is #pir^2*h_2/3#
where #h_2# is the height of the cone.

The volume of the solid is the sum of those two volumes: #V=pir^2*h_1 + pir^2*h_2/3#
Factoring #pir^2#:
#V= pir^2(h_1+h_2/3)#
#64pi = pir^2(5+66/3) = pir^2(5+22) = pir^2(27)#

#64pi = 27pir^2#

#pir^2 = (64pi)/27#
And that is the area of the base: #A=pir^2=(64pi)/27#