Question

A cone has a height of `12 cm` and its base has a radius of `6 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

`color(green)(318.24` `color(green)(cm^2`

Explanation:

The bottom segment shown in the diagram is called a `"frustum"`

We need to find the surface area of it. We use the formula

We know that `R=6` and `h=5`

But we should find the value of `r`

`r` is also the radius of base of small cone

We can use the ratios of the height's to solve it

`color(orange)(("Radius of original cone")/("Height of original cone")=("Radius of smaller cone")/("Height of smaller cone")`

`rarr6/12=r/7`

`rarr1/2=r/7`

`rarrr=7*1/2`

`color(purple)(rArrr=3.5`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let's make clear of all the variables

`color(indigo)(R=6`

`color(indigo)(h=5`

`color(indigo)(r=3.5`

`rarrpi(R+r)sqrt((R-r)^2+h^2)+pir^2+piR^2`

`rarr22/7(6+3.5)sqrt((6-3.5)^2+5^2)+22/7*3.5^2+22/7*6^2`

`rarr22/7*9.5sqrt((2.5)^2+25)+3.14*12.25+3.14*36`

`rarr3.14*9.5sqrt(6.25+25^color(white)(2))+38.46+113.04`

`rarr29.83sqrt(31.25^color(white)(2))+151.5`

`rarr29.83*5.59+151.5`

`rarr166.74+151.5`

`color(green)(rArr318.24` `color(green)(cm^2`