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?

1 Answer

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.