Question

A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in figure. The height of the entire rocket is 26cm, while the height of the conical part is 6cm. The base of the conical portion has a diameter of 5cm, while the base diameter of the cylindrical portion is 3cm. If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colors. (Take π=3.14)

Answer 1

Hint: - There is a little area at the bottom of the cone which is to be painted orange and that would be calculated by just removing the area of the base of the cylinder from that of the cone.
The curved surface area of a cylinder is
\[CSA=2\pi rh\]
The curved surface area of the cone is
\[CSA=\pi rl\]
Area of the base of a cylinder or a cone is
\[Area=\pi {{r}^{2}}\]

Complete step-by-step answer:
As mentioned in the question,
Area of the rocket to be painted orange = Curved surface area of the cone + Base area of the cone − Base area of the cylinder
Area of the rocket to be painted yellow = Curved surface area of the cylinder + Area of one of the bases of the cylinder
 Curved surface area of cone can be calculated by using formula given in the hint as
\[\pi rl\] .
(Where ‘l’ is the slant height of the cone)
Now, the slant height l can be evaluated as
\[\begin{align}
  & {{l}^{2}}={{r}^{2}}+{{h}^{2}} \\
 & {{l}^{2}}={{6}^{2}}+{{2.5}^{2}} \\
 & {{l}^{2}}=36+6.25 \\
 & {{l}^{2}}=42.25 \\
 & l=6.5\ cm \\
\end{align}\] \[\]
Hence, the curved surface of the cone is
\[\begin{align}
  & =\pi rl \\
 & =3.14\times 2.5\times 6.5 \\
 & =51.025\ c{{m}^{2}} \\
\end{align}\]
Base area of cone can be calculated as
\[\begin{array}{*{35}{l}}
   =\pi {{r}^{2}} \\
   =3.14\times {{\left( 2.5 \right)}^{2}} \\
   =3.14\times 6.25 \\
   =19.625~\ c{{m}^{2}} \\
\end{array}\]
Curved surface area of cylinder is given by using the formula as mentioned in the hint as
\[\begin{align}
  & =2\pi (\text {radius of the cylinder})h \\
 & \begin{array}{*{35}{l}}
   =2\times 3.14\times 1.5\times 20 \\
   =188.4~\ c{{m}^{2}} \\
\end{array} \\
\end{align}\]
Base area of cone can be calculated as
\[\begin{array}{*{35}{l}}
   ~=\pi r^2 \\
   =3.14\times {{\left( 1.5 \right)}^{2}} \\
   =7.065~\ c{{m}^{2}} \\
\end{array}\]
Area to be painted orange = Curved surface area of the cone + Base area of the cone − Base area of the cylinder
\[\begin{array}{*{35}{l}}
   =51.025+19.625-7.0625 \\
   =63.59~c{{m}^{2}} \\
   ~ \\
\end{array}\]
Area to be painted yellow = Curved surface area of the cylinder + Area of one bottom base of the cylinder
\[\begin{array}{*{35}{l}}
   =188.4+7.065 \\
   =195.465~\ c{{m}^{2}} \\
\end{array}\]
 Hence,
\[\begin{array}{*{35}{l}}
   Area\text{ }to\text{ }be\text{ }painted\text{ }orange~=63.59~\ c{{m}^{2}} \\
   Area\text{ }to\text{ }be\text{ }painted\text{ }yellow~=195.465~\ c{{m}^{2}} \\
\end{array}\]

Note: - The students can make an error in considering the little area at the bottom of the cone which is to be painted orange and that would be calculated by just removing the area of the base of the cylinder from that of the cone.