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?

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 #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?

Maverick Smith · Accepted Answer

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

Sadie Allen · Answer

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

James Atkins · Answer

undefined The area of the base of the cylinder is 16 square units.