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 #12 #. If the volume of the solid is #24 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 #12   #.  If the volume of the solid is #24 pi#, what is the area of the base of the cylinder?

Aaliyah Anderson · Accepted Answer

#8/3 pi#

Consider the solid

We can say that the volume of the solid equals the sum of the volume of the cone and the cylinder

#color(blue)(V_(cy)+V_(co)=24pi#

Volume of cylinder#=color(purple)(pir^2h#

Volume of cone#=color(indigo)(1/3pir^2h#
(#1/3# of the volume of cylinder)

Area of the base (the base is a circle)#=color(orange)(pir^2#

#n##ot##e##:pi=22/7#

If you take a closer look at the formula, you could see #pir^2# appearing in both. So,let #pir^2# be #w#

#color(orange)(pir^2=w#

#rarrwh+1/3wh=24pi#

#rarrw*12+1/3*w*9=24pi#

#rarrw*12+1/cancel3^1*w*cancel9^3=24pi#

#rarr12w+3w=25pi#

#rarr15w=24pi#

#color(green)(rArrw=(24pi)/15=8/3pi#

(area of the base)

Evelyn Arboleda · Answer

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

Given that the height of the cone is $ 9 $ and the height of the cylinder is $ 12 $, we can use the formula for the volume of a cone and a cylinder to set up an equation.

The volume $ V $ of the solid is given as $ 24\pi $, so:

$$ V = \text{Volume of Cone} + \text{Volume of Cylinder} $$

$$ 24\pi = \frac{1}{3} \pi r^2 \cdot 9 + \pi r^2 \cdot 12 $$

We can simplify this equation:

$$ 24\pi = 3\pi r^2 + 12\pi r^2 $$

$$ 24\pi = 15\pi r^2 $$

Now, solve for $ r^2 $:

$$ r^2 = \frac{24\pi}{15\pi} $$

$$ r^2 = \frac{8}{5} $$

$$ r = \sqrt{\frac{8}{5}} $$

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

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

$$ A = \pi r^2 $$

$$ A = \pi \left(\frac{2\sqrt{5}}{5}\right)^2 $$

$$ A = \pi \cdot \frac{4 \cdot 5}{25} $$

$$ A = \frac{20\pi}{25} $$

$$ A = \frac{4\pi}{5} $$

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