A cone has a height of #5 cm# and its base has a radius of #4 cm#. If the cone is horizontally cut into two segments #2 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #5  cm# and its base has a radius of #4  cm#.  If the cone is horizontally cut into two segments #2 cm# from the base, what would the surface area of the bottom segment be?

Avery Allen · Accepted Answer

#119.858\ ext{cm}^2# Radius #r# of new circular section of bottom segment cut horizontally, at a height #h=2\ cm# from base, from a original cone of height #H=5\ cm# & base radius #R=4\ cm#, is given by using property of similar triangles as follows #\frac{R-r}{h}=\frac{R}{H}# #r=R(1-\frac{h}{H})# #=4(1-2/5)# #=2.4\ cm# Now, surface area of bottom segment of original cone #= ext{area of circular top of radius 2.4 cm}+ ext{curved surface area of frustum of cone}+ ext{area of circular base of radius 4 cm}# #=\pir^2+\pi(r+R)\sqrt{h^2+(R-r)^2}+\piR^2# #=\pi(2.4)^2+\pi(2.4+4)\sqrt{2^2+(4-2.4)^2}+\pi(4)^2# #=119.858\ ext{cm}^2#

Mason Adams · Answer

undefined To find the surface area of the bottom segment of the cone, we first need to calculate the slant height of the cone's frustum, which is the segment left after cutting the cone horizontally.

Using the Pythagorean theorem, the slant height (l) of the frustum can be calculated as follows:

l = √(r^2 + h^2)

Where r is the radius of the base and h is the height of the frustum.

Given that the radius (r) of the base is 4 cm and the height (h) of the frustum is 5 - 2 = 3 cm (since the cone is cut 2 cm from the base), we can calculate:

l = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm

Now, to find the surface area of the bottom segment of the cone (the frustum), we use the formula:

Surface area = π(r1 + r2)l

Where r1 and r2 are the radii of the base and top of the frustum respectively, and l is the slant height.

Since the frustum is cut 2 cm from the base, the radius of the top of the frustum is 2 cm less than the radius of the base. Therefore, r2 = 4 - 2 = 2 cm.

Substituting the values into the formula, we get:

Surface area = π(4 + 2)5 = π(6)(5) = 30π cm^2

So, the surface area of the bottom segment of the cone is 30π square centimeters.