Question

In the given figure, from the top of a solid cone of height 12 cm and base radius 6cm, 3cone of height 4cm is removed by a plane parallel to the base. Find the total surface of the remaining solid. $ \left[ {use\pi = \dfrac{{22}}{7}and\sqrt 5 = 2.236} \right] $

Answer 1

Hint: First try to understand the shape of the remaining part of the cone when the upper part is removed. In this case the remaining part is known as the frustum. It is simplified now as we can use the formula of frustum to find it’s surface area. Use the given measurements and calculate the surface area.

Complete step-by-step answer:
Since the cone of height 4cm is removed from the top of the main cone, the part that is remaining is called the frustum.
The frustum has a top circle, bottom circle, and circular height.

We have to find the total surface area of this frustum.
Surface area= Area of top circle + area of bottom circle + Area the circular height
 $ = \pi {r_1}^2 + \pi {r_2}^2 + \pi l\left( {{r_1} + {r_2}} \right) $
where, $ {r_1} $ = radius of top circle.
 $ {r_2} $ = radius of the bottom circle.
 $ l $ = slant height of the frustum which is given by $ l = \sqrt {{h^2} + {{\left( {{r_2} - {r_1}} \right)}^2}} $
where h= height of the frustum.
We will first try to find the slant height of the frustum
 $ {r_1} $ = 2cm
 $ {r_2} $ =6cm
h=total height of the cone-height of the cone removed
=12-4
=8
 $
  l = \sqrt {{h^2} + {{\left( {{r_2} - {r_1}} \right)}^2}} \\
   = \sqrt {{8^2} + {{\left( {6 - 2} \right)}^2}} \\
   = \sqrt {64 + 16} \\
   = 4\sqrt 5 \;
  $
Hence, we will substitute all this value to find total surface area of frustum
Surface area $ = \pi {r_1}^2 + \pi {r_2}^2 + \pi l\left( {{r_1} + {r_2}} \right) $
 $
   = \left( {\dfrac{{22}}{7} \times {2^2}} \right) + \left( {\dfrac{{22}}{7} \times {6^2}} \right) + \left( {\dfrac{{22}}{7} \times 4\sqrt 5 \left( {2 + 6} \right)} \right) \\
   = \dfrac{{88}}{7} + \dfrac{{792}}{7} + 224.8777 \\
   = 350.591 \;
  $
Hence the total surface area of the remaining part is 350.591 $ c{m^2} $
So, the correct answer is “350.591 $ c{m^2} $ ”.

Note: Whenever this type of question arrives where we have been told to find the area of the surface or volume or area after deletion of a certain part, first try to find the shape of the newly formed. This makes the process easy as we can determine its formula and hence make the process easier to find our aim.