A solid consists of a cone on top of a cylinder with a radius equal to the cone. The height of the cone is #3 # and the height of the cylinder is #5 #. 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 #3  # and the height of the cylinder is #5   #.  If the volume of the solid is #20 pi#, what is the area of the base of cylinder?

Sadie Adams · Accepted Answer

The area of the base is #=10.5u^2#

Let #a=# area of the base

Volume of cone is #V_(co)=1/3*a*h_(co)#

Volume of cylinder is #V_(cy)=a*h_(cy)#

Total amount

#V=V_(co)+V_(cy)#

#V=1/3ah_(co)+ah_(cy)#

#20pi=a(1/3*3+5)#

#20pi=a*6#

#a=20/6pi=10/3pi#

#a=10.5#

Claire Akin · Answer

undefined To find the area of the base of the cylinder, we need to first determine the radius of both the cone and the cylinder. Since the radius of the cylinder is equal to the radius of the cone, let's denote this radius as $ r $.

The volume of the solid is given as 20π. We can express the volume of the solid as the sum of the volumes of the cone and the cylinder:

$$ V_{\text{solid}} = V_{\text{cone}} + V_{\text{cylinder}} $$

Given that the height of the cone is 3 and the height of the cylinder is 5, and using the formula for the volumes of a cone and a cylinder:

$$ V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}} $$
$$ V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}} $$

We can substitute these into the equation for the volume of the solid:

$$ 20\pi = \frac{1}{3} \pi r^2 (3) + \pi r^2 (5) $$

Simplify the equation:

$$ 20\pi = \pi r^2 (1 + 15) $$
$$ 20\pi = 16\pi r^2 $$

Now, divide both sides by $ 16\pi $ to solve for $ r^2 $:

$$ r^2 = \frac{20\pi}{16\pi} $$
$$ r^2 = \frac{5}{4} $$

Taking the square root of both sides:

$$ r = \frac{\sqrt{5}}{2} $$

Now that we have found the radius of the cylinder, we can calculate the area of the base of the cylinder using the formula for the area of a circle:

$$ A_{\text{base}} = \pi r^2 $$

Substituting the value of $ r $:

$$ A_{\text{base}} = \pi \left(\frac{\sqrt{5}}{2}\right)^2 $$
$$ A_{\text{base}} = \pi \left(\frac{5}{4}\right) $$
$$ A_{\text{base}} = \frac{5\pi}{4} $$

So, the area of the base of the cylinder is $ \frac{5\pi}{4} $.