Question

A cone has a height of `6 cm` and its base has a radius of `8 cm`. If the cone is horizontally cut into two segments `5 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

Surface Area of the bottom segment `color(blue)(S_b=(700pi)/9" ")`square units

Explanation:

After the cone has been cut horizontally 5 cm from the base, there will be a smaller cone on top with a height of 1 cm. We need to determine the radius `r_t` of this smaller cone.

By ratio and proportion

`r_t/1=8/6`

`r_t=4/3" "`cm

We now calculate the surface area `S_t` of the smaller cone

`S_t=pi *r_t*sqrt(1^2+(4/3)^2)" "`where `sqrt(1^2+(4/3)^2)` is the slant height of the smaller cone.

`S_t=pi*(4/3)*sqrt(1^2+(4/3)^2)`

`S_t=pi*(4/3)*(5/3)`

`S_t=(20pi)/9`

Compute now the surface area `S_b` of the original cone by subtraction

`S_b=S_o-S_t`

`S_b=pi*r*l-(20pi)/9`

`S_b=pi(8)(10)-(20pi)/9`

`S_b=((720pi)-(20pi))/9`

`color(blue)(S_b=(700pi)/9" ")`square units

God bless....I hope the explanation is useful.