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?

Answer 1

TSA = `= 408.55`

Explanation:

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.