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 #60 # and the height of the cylinder is #15 #. If the volume of the solid is #7 pi#, what is the area of the base of the cylinder?

1 Answer
Oct 23, 2016

The area of the base of the cylinder is #(7pi)/100#.

Explanation:

The volume of the solid is found by adding the volume of the cylinder to the volume of the cone. The volume of a cylinder is given by #V = pir^2h#. The volume of the cone is given by #V = (pir^2h)/3#. So the volume of the entire solid is given by:

#V = pir^2h + (pir^2h)/3#

This can be transformed by the converse of the Distributive Property to become:

#V = pir^2h(1 + 1/3)#

Which then becomes:

#V = (4pir^2h)/3#

Since we are looking for the area of the base, which is a circle, we need to isolate the part of the formula which gives that area. The area of a circle is given by #A = pir^2#, so we need to isolate that part of the simplified volume formula.

#3*V = 3*(4pir^2h)/3#
#3V = 4pir^2h#
#(3V)/(4h) = (4pir^2h)/(4h)#
#(3V)/(4h) = pir^2#

Therefore, the area of the base of the cylinder is given by the formula:

#A = (3V)/(4h)#

Substitute the values you know for #V# and #h# and simplify as possible. {#h = 60 + 15 = 75#}

#A = (3*7pi)/(4*75)#
#A = (21pi)/300#
#A = (7pi)/100#

Since the volume in the problem is left in terms of #pi#, it is acceptable to give the area of the base in terms of #pi# as well.