Question

A cone has a height of `18 cm` and its base has a radius of `4 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

`:.color(purple)(=172.22cm^2` to the nearest 2 decimal places cm^2#

Explanation:

:.Pythagoras: `c^2=18^2+4^2`

`:.c=L=sqrt(18^2+4^2)`

`:. c=Lcolor(purple)(=18.439cm`

`:.18/4=tan theta=4.5=77^@28’16.2”`

:.`color(purple)(S.A.`= pir^2+pir*L#

:.S.A.`=pi*4*18.439`

:.S.A.`=231.711`

:.Total S.A.`color(purple)(=262.116cm^2`

`:.Cot 77^@28’16.2”*14=3.111cm=`radius of top part

:.Pythagoras: `c^2=14^2+3.111^2`

`:.c=L=sqrt(14^2+3.111^2)`

`:. c=Lcolor(purple)(=14.341cm` top part
:.S.A. top part`=pi*r*L`

S.A. top part`:.pi*3.111*14.341`

S.A. top part`:.=140.162`

S.A. top part`:.color(purple)(=140.162cm^2`

:.S.A. Bottom part`color(purple)(=231.711-140.162=91.549cm^2`

:.S.A. Bottom part`=91.549+30.408+50.265=172.222 cm^2`

`:.color(purple)(=172.22cm^2` to the nearest 2 decimal places `cm^2`

Answer 2

`:.color(purple)(=172.22cm^2` to the nearest 2 decimal places `cm^2`

Explanation:

:.Pythagoras: `c^2=18^2+4^2`

`:.c=L=sqrt(18^2+4^2)`

`:. c=Lcolor(purple)(=18.439cm`
`:.18/4=tan theta=4.5=77^@28’16.2”`

:.`color(purple)(S.A.`=pirL#

:.S.A.`=pi*4*18.439`

:.S.A.`=231.711`

:.Total S.A.`color(purple)(=231.711cm^2`

`:.Cot 77^@28’16.2”*14=3.111cm=`radius of top part

:.Pythagoras: `c^2=14^2+3.111^2`

`:.c=L=sqrt(14^2+3.111^2)`

`:. c=Lcolor(purple)(=14.341cm` top part
:.S.A. top part`=pi*r^2+pi*r*L`

S.A. top part`:.pi*3.111*14.341`

S.A. top part`:.=140.162`

S.A. top part`:.color(purple)(=140162cm^2`

:.S.A. Bottom part`color(purple)(=231.711-140.162=91.549cm^2`

`91.549+30.408+50.265=172.222`

`:.color(purple)(=172.22cm^2` to the nearest 2 decimal places `cm^2`