Question

A cone has a height of `12 cm` and its base has a radius of `2 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?

Answer 1

`color(blue)(77.92 "cm"^2)` 2 d.p.

Explanation:

If we start with a cone of height 12cm and cut the top off, the height of the top section will be:

`12-7=5cm` This has been reduced by a factor of `5/12`

The radius of the top section (which is still a cone ) will also be reduced by the same factor:

`5/12*2=5/6`

We will call these:

`r=5/6` and `h=5`

The formula for the lateral surface area of a cone is given as:

`A=pirsqrt(h^2+r^2)`

Note:

The lateral surface area is the area of the cone minus the base area of the cone.

Let `R="radius of given cone"`

Let `H="height of given cone"`

If we now find the lateral surface area of the given cone and subtract the lateral surface area of the top cone, we will be left with the lateral surface area of the bottom section. Then we can add the surface area of the base and the top of this section( this will have the same radius as the top cone section):

We can combine this into one formula:

`piRsqrt(H^2+R^2)-(pirsqrt(h^2+r^2))+piR^2+pir^2`

`pi(Rsqrt(H^2+R^2)-rsqrt(h^2+r^2)+R^2+r^2)`

Plugging in `R=2` , `H=12`, `r=5/6`, `h=5`

`:.`

`pi(2sqrt(12^2+2^2)-5/6sqrt(5^2+(5/6)^2)+2^2+(5/6)^2)`

`pi((119sqrt(37))/36+169/36)=77.92cm^2` 2 .d.p.