A cone has a height of #16 cm# and its base has a radius of #3 cm#. If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

By cutting off a segment of a cone parallel to the base, you create what is known as a frustum. There are four important pieces of information in a fustrum: The height #(h)#, the larger radius #(R_1)#, the smaller radius #(R_2)# and the slant #(s)#. In the question asked we know two of these variables, h=7 and R1=3. The first thing we need to do is find #R_2#.

By looking at the question, we see the original cone has a height of 16cm and a radius of 3cm. This means the relationship between the height and the radius is equal to #16/3#. In the frustum, we have the height, but the smaller radius is unknown, which makes the relationship of height to radius equal to #7/R_2#. Since the ratios of the cone have been unchanged while making it a fustrum, we can safely say that the height-radius ratio of the cone is the same in the fustrum, so
#16/3=7/R_2#. By cross-multiplying, we find that
#21=16R_2#. Divide both sides by 16, and we get
#21/16=R_2#
#1.3125=R_2#

Now we have the values of #h, R_1 and R_2#, all that is left is #s#. The formula for finding #s# is as follows:
#s=sqrt((R_1-R_2)^2+h^2)#
By looking at the image below, you should get an understanding of how this works using Pythagoras' Theorem.
Lastly, we plug in all our known values into the equation to get
#A_s=pi(sqrt(51.8476)(3-1.3125)+3^2+(1.3125)^2)#
#A_s=pi(12.151+9+1.7226)#
#A_s=pi(22.874)#
#A_s~~71.859cm^2#
Therefore the surface area of the bottom segment of the cone is roughly 71.859#cm^2#.

I hope I helped