A cone has a height of #15 cm# and its base has a radius of #8 cm#. If the cone is horizontally cut into two segments #8 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #15  cm# and its base has a radius of #8  cm#.  If the cone is horizontally cut into two segments #8 cm# from the base, what would the surface area of the bottom segment be?

Evelyn Arboleda · Accepted Answer

#579.047\ ext{cm}^2# Radius #r# of new circular section of bottom segment cut horizontally, at a height #h=8\ cm# from base, from a original cone of height #H=15\ cm# & base radius #R=8\ cm# by using property of similar triangle we get #\frac{R-r}{h}=\frac{R}{H}# #r=R(1-\frac{h}{H})# #=8(1-8/15)# #=3.733\ cm# Now, surface area of bottom segment of original cone #= ext{area of circular top of radius 3.733}+ ext{curved surface area of frustum of cone}+ ext{area of circular base of radius 8}# #=\pir^2+\pi(r+R)\sqrt{h^2+(R-r)^2}+\piR^2# #=\pi(3.733)^2+\pi(3.733+8)\sqrt{8^2+(8-3.733)^2}+\pi(8)^2# #=579.047\ ext{cm}^2#

Theodore Abad · Answer

undefined To find the surface area of the bottom segment of the cone after it's horizontally cut 8 cm from the base, you need to calculate the surface area of the smaller cone formed by this segment.

The smaller cone's height would be 15 cm - 8 cm = 7 cm, and its radius would remain 8 cm.

Using the formula for the surface area of a cone (A = πr^2 + πrl, where r is the radius and l is the slant height), you can calculate the surface area of the bottom segment.

First, find the slant height (l) of the smaller cone using the Pythagorean theorem: l = √(r^2 + h^2) = √(8^2 + 7^2) ≈ 10.63 cm.

Then, use the surface area formula: A = π(8^2) + π(8)(10.63) ≈ 201.06 cm^2.

So, the surface area of the bottom segment of the cone is approximately 201.06 square centimeters.