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 #12 # and the height of the cylinder is #28 #. If the volume of the solid is #42 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 #12  # and the height of the cylinder is #28   #.  If the volume of the solid is #42 pi#, what is the area of the base of the cylinder?

Theodore Abad · Accepted Answer

#(21pi)/16#

Assume the radius of the cylinder/cone as r, height of cone as #h_1#, height of cylinder as #h_2#

Volume of the cone part of solid = #(pi*r^2*h_1)/3#

Volume of the cylinder part of solid = # pi*r^2 * h_2#

What we currently possess is:

#h_1# = 12,#h_2# = 28

#(pi*r^2*h_1)/3# + # pi*r^2 * h_2# = #42*pi#

#(pi*r^2*12)/3# + # pi*r^2 * 28# = #42*pi#

# pi*r^2 * 4# + # pi*r^2 * 28# = #42*pi#

# 32pi*r^2 # = #42*pi#

# r^2 # = #42/32# = #21/16#

Area of the base of the cylinder = #pi*r^2# = #pi*21/16# = #(21pi)/16#

Savannah Aguilar · Answer

undefined The area of the base of the cylinder can be found by first determining the volume of the cone and the cylinder, then subtracting their combined volume from the total volume of the solid, and finally solving for the area of the base of the cylinder using the remaining volume.

Given that the volume of the solid is $42\pi$, and the heights of the cone and cylinder are $12$ and $28$ respectively, the volume of the cone is $\frac{1}{3}\pi r^2 h$ and the volume of the cylinder is $\pi r^2 h$, where $r$ is the radius and $h$ is the height.

Thus, we have:

Volume of cone: $\frac{1}{3}\pi r^2 \times 12 = 4\pi r^2$
Volume of cylinder: $\pi r^2 \times 28 = 28\pi r^2$

The total volume of the cone and cylinder combined is $4\pi r^2 + 28\pi r^2 = 32\pi r^2$.

Therefore, the volume of the base of the cylinder is $42\pi - 32\pi r^2 = 10\pi r^2$.

We can solve for $r$ using the given total volume of the solid:

$$ 10\pi r^2 = 42\pi $$
$$ r^2 = \frac{42\pi}{10\pi} = \frac{21}{5} $$

Now, we can find the radius $r$:

$$ r = \sqrt{\frac{21}{5}} = \frac{\sqrt{21}}{\sqrt{5}} = \frac{\sqrt{21} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{105}}{5} $$

The area of the base of the cylinder is $ \pi r^2 = \pi \left( \frac{\sqrt{105}}{5} \right)^2 = \pi \frac{105}{25} = \frac{21\pi}{5} $