A solid consists of a cone on top of a cylinder with a radius equal to the cone. The height of the cone is #6 # and the height of the cylinder is #8 #. If the volume of the solid is #20 pi#, what is the area of the base of cylinder?

Question

A solid consists of a cone on top of a cylinder with a radius equal to the cone.  The height of the cone is #6  # and the height of the cylinder is #8   #.  If the volume of the solid is #20 pi#, what is the area of the base of cylinder?

Jayden Jackson · Accepted Answer

#pir^2=SA=(30pi)/13#

In order to make things easier to follow, I'm going to use the letters N and L to stand for Cone and Cylinder, respectively.

We know:

#h_N=6#

#h_L=8#

#r_L=r_N# and so I can refer to both as simply #r#

#V_N+V_L=20pi#

What is the size of the cylinder's base?

First, let's enter the values for the volumes of the two shapes into the formulas:

#V_N=pir^2h#

#V_L=(pir^2h)/3#

#pir^2h_N+(pir^2h_L)/3=20pi# and now substitute in values:

#pir^2(6)+(pir^2(8))/3=20pi#

#(pir^2(18))/3+(pir^2(8))/3=20pi#

#(pir^2(26))/3=20pi#

#(r^2(26))/3=20#

#r^2(26)=60#

#r^2=60/26=30/13#

The area of the base is #SA=pir^2# and so:

#pir^2=SA=(30pi)/13#

Aaliyah Anderson · Answer

undefined Given that the solid consists of a cone on top of a cylinder with the same radius, and the volume of the solid is 20π, and the height of the cone is 6 and the height of the cylinder is 8, we can find the area of the base of the cylinder.

The volume of the solid is the sum of the volumes of the cone and the cylinder:

Volume of cone = (1/3)πr^2h
Volume of cylinder = πr^2h

Given that the volume of the solid is 20π and the height of the cone is 6, we can set up the equation:

(1/3)πr^2 * 6 + πr^2 * 8 = 20π

Solving this equation for r:

2πr^2 + 6πr^2 = 20π
8πr^2 = 20π
r^2 = 20/8
r^2 = 5/2
r = √(5/2)

Now that we have the radius of the cylinder, we can find the area of the base of the cylinder:

Area of base of cylinder = πr^2
Area of base of cylinder = π * (5/2)
Area of base of cylinder = (5π)/2

Therefore, the area of the base of the cylinder is (5π)/2 square units.