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 #11 #. If the volume of the solid is #140 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 #11   #.  If the volume of the solid is #140 pi#, what is the area of the base of the cylinder?

Caroline Aucoin · Accepted Answer

The formula for volume of a cone is #1/3A_(base) xx h# while the formula for volume of a cylinder is #A_(base) xx h#. The area of the base is the formula for the area of a circle, so #pi xx r^2#

The radius, r, must be determined before we can begin our search for the area of the base.

#1/3 xx 9 xx (r^2 xx pi) + (r^2 xx pi) xx 11 = 140pi#

#3pir^2 + 11pir^2 = 140pi#

#pi(3r^2 + 11r^2) = 140pi#

#14r^2 = (140pi)/pi#

#r^2 = 10#

#r = sqrt(10)#

#A = r^2 xx pi#

#A = (sqrt(10))^2 xx pi#

#A = 10pi#

The area of the base of the cylinder measures #10pi# square units.

I hope this is useful!

Carson Arnold · Answer

undefined The volume $ V $ of the solid consisting of a cone on top of a cylinder is given by $ V = \frac{1}{3} \pi r^2 h_{\text{cone}} + \pi r^2 h_{\text{cylinder}} $, where $ r $ is the radius, $ h_{\text{cone}} $ is the height of the cone, and $ h_{\text{cylinder}} $ is the height of the cylinder. Substituting the given values and solving for $ r $, we find $ r = 3 $. Then, to find the area of the base of the cylinder, we use the formula for the area $ A $ of a circle: $ A = \pi r^2 $. Substituting $ r = 3 $, we find $ A = 9\pi $. Therefore, the area of the base of the cylinder is $ 9\pi $.