Question

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Answer 1

Hint:
Find the curved surface area of the cone and the hemisphere respectively and then add them up to get the total surface area of the toy.
Curved surface area of the hemisphere\[ = 2\pi {r^2}\]
Curved surface area of the cone\[ = \pi rl\]

Complete step by step solution:
Given the toy is in the form of a cone mounted over a hemisphere as shown in the diagram below, hence to find the total surface area first find the curved surface area of the hemisphere and then the cone and then we will add them up

Curved surface area of the hemisphere:

Given radius \[r = 3.5cm\]
Curved surface area of the hemisphere is given by the formula \[ = 2\pi {r^2} - - (i)\]
Hence by substituting the value of r in the formula, we get
\[
  CS{A_1} = 2\pi {r^2} \\
   = 2 \times \dfrac{{22}}{7} \times 3.5 \times 3.5 \\
   = 22 \times 3.5 \\
   = 77c{m^2} \\
 \]
Curved surface area of the cone:

Curved surface area of the cone is given by the formula \[ = \pi rl - - (ii)\]
Where, the radius of the cone \[r = 3.5cm\]
Height of cone (h) = height of toy –Height of hemisphere\[ = 15.5 - 3.5 = 12cm\]
Now for slant height of the cone, by using Pythagoras theorem, we can write
\[{l^2} = {h^2} + {r^2} - - (iii)\]
Hence by substituting the value of height and radius, we get
\[
  {l^2} = {h^2} + {r^2} \\
  {l^2} = {\left( {12} \right)^2} + {\left( {3.5} \right)^2} \\
  l = \sqrt {144 + 12.25} \\
  l = \sqrt {156.25} \\
  l = 12.5cm \\
 \]
Hence the curved surface area of the cone will be
\[
  CS{A_2} = \pi rl \\
   = \dfrac{{22}}{7} \times 3.5 \times 12.5 \\
   = 11 \times 12.5 \\
   = 137.5c{m^2} \\
 \]
Therefore the total surface area of the toy which is the sum of the curved surface area of a cone and the hemisphere will be
\[
  TSA = CS{A_1} + CS{A_2} \\
   = 77 + 137.5 \\
   = 214.5c{m^2} \\
 \]

Therefore the total surface area of the toy \[ = 214.5c{m^2}\]

Note:
Students should note that whenever they are given a composite figure to find their area, volume, surface area, they must separate that figure in regular geometrical shapes the find their area, surface area and then they must add them up.