Question

A fez, the cap used by Turks, is shaped like the frustum of a cone. If its radius on the open side is 10 cm , radius at the upper base is 4cm and its slant height is 15 cm, find the area of material used for making it.

Answer 1

Hint- The area of material used to make the cup will be equal to the curved surface area of frustum and the area of circle portion of upper side, use this formula to get the answer.
Complete Step-by-Step solution:
Since the fez is in the form of frustum and is covered from the upper side.
Area of material used to make cap = curved surface area of frustum $ + $ area of circle portion of upper side
$ = \pi \left( {{r_1} + {r_2}} \right)l + \pi {\left( {{r_2}} \right)^2}$
Here
\[
  {r_1} = {\text{Radius of lower base = 10cm}} \\
  {r_2} = {\text{Radius of upper base = 4 cm}} \\
  l = {\text{Slant height = 15 cm}} \\
\]
Therefore the area of the material is given by
$
   = \pi \left( {{r_1} + {r_2}} \right)l + \pi {\left( {{r_2}} \right)^2} \\
   = \pi \left( {4 + 10} \right) \times 15 + \pi {\left( 4 \right)^2} \\
   = \pi \times 14 \times 15 + \pi \times 4 \times 4 \\
   = \pi \times 210 + \pi \times 16 \\
   = \pi \times 226 \\
   = \dfrac{{22}}{7} \times 226 \\
   = \dfrac{{4972}}{7} = 710\dfrac{2}{7}{\text{c}}{{\text{m}}^2} \\
 $
Hence, area of material is $710\dfrac{2}{7}{\text{c}}{{\text{m}}^2}$ .

Note- In order to solve these types of questions, remember the formula of area and curved surface area of cone, circle, cube, rhombus etc. Drawing the diagram of the given shape first, this helps us to know how to approach the problem and which area we need to calculate. Some problems may consist of sectioning of the cone, sphere etc. so drawing the diagram of the problem helps a lot.