A cone has a height of #16 cm# and its base has a radius of #7 cm#. If the cone is horizontally cut into two segments #4 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #16  cm# and its base has a radius of #7  cm#.  If the cone is horizontally cut into two segments #4 cm# from the base, what would the surface area of the bottom segment be?

Sadie Allen · Accepted Answer

TSA = #= 408.55#

If the cone is cut horizontally, the cone which is formed is similar to the original cone. This means that all the lengths are in the same ratio.

We can therefore find the radius of the smaller cone using direct proportion and comparing the heights with the radii.

The height of the smaller (removed) cone will be 12cm.

To find the radius of the base of the smaller cone:

#r/7 = 12/16 rArr r = (7 xx 12) /16 rArr r = 5.25 cm#

The curved surface area of the original cone is given by: TSA = #pi R l#

#= pi xx 7 xx sqrt305#

The curved surface area of the smaller removed cone is

#= pi xx 5.25 xx sqrt171.5625"

Therefore the curved area of the remaining bottom segment is: #= (pi xx 7 xx sqrt305) - (pi xx 5.25 xx sqrt171.5625)#

The area of the two circular surfaces need to be added to this, #pi 7^2 + pi5.25^2#

TSA = #= (pi xx 7 xx sqrt305) - (pi xx 5.25 xx sqrt171.5625) + pi 7^2 + pi5.25^2 #

This can be written as #pi( 7sqrt305 - 5.25sqrt171.5625 + 7^2 + 5.25^2)#

I am not convinced that working with the areas of the similar figures would be any easier.

James Atkins · Answer

undefined To find the surface area of the bottom segment of the cone after it is cut horizontally 4 cm from the base, follow these steps:

1. First, calculate the slant height ($l$) of the cone using the Pythagorean theorem:
   $$ l = \sqrt{h^2 + r^2} $$
   where $h$ is the height of the cone and $r$ is the radius of its base.
   Substituting the given values: $h = 16$ cm and $r = 7$ cm.

2. Once you have the slant height, you can find the radius of the smaller cone formed by the cut. 
   $$ r_2 = \frac{h - 4}{h} \times r $$
   where $r_2$ is the radius of the smaller cone.

3. Now, you can calculate the surface area of the bottom segment of the original cone using the formula:
   $$ A = \pi r_2 (r + r_2) $$
   where $A$ is the surface area of the bottom segment.

Substitute the values you've found into the equations to get the final answer.