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?

1 Answer
Oct 11, 2016

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

Explanation:

enter image source here

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

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

enter image source here
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#

enter image source here

#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#