Question

A circular tent is in the form of a cone over a cylinder. The diameter of the base is 9m, the height of the cylindrical part is 4.8m and the total height of the tent is 10.8m. The canvas required for the tent is______
A.24.184 sq. m
B.2418.4 sq. m
C.241.84 sq. m
D.None of these.

Answer 1

Hint: To solve this first we need to find the curved surface area of the cylinder and we find the curved surface area of the cone. By adding curved surface area of cylinder and cone we get an area of canvas. Given diameter of cylinder and height of a cylinder. The total height of the tent is given, subtracting with height of a cylinder we get height of a cone.

Complete step-by-step answer:
We know, curved surface area of the cylinder is \[ \Rightarrow 2\pi rh\] .
Height of a cylinder is \[ \Rightarrow h = 4.8\;m\] .
Diameter of a cylinder is \[ \Rightarrow d = 9\;m\] .
Then, radius \[ \Rightarrow r = \dfrac{d}{2} = \dfrac{9}{2} = 4.5m\] .
Curved surface area of cylinder \[ = 2\pi rh\]
 \[ = 2 \times \pi \times 4.5 \times 4.8\]
 \[ = 2 \times \dfrac{{22}}{7} \times 21.6\]
 \[ = \dfrac{{950.4}}{7}\]
 \[ = 135.77\;{m^2}\] .
  

Curved surface area of cone is \[ = \pi rl\]
As we can see in the figure the radius is the same for cylinder and cone. Slant height is given by \[l = \sqrt {{h^2} + {r^2}} \] , here \[h\] is the height of the cone. Height of cone=total height of tent- height of cylinder.
That is \[h = 10.8 - 4.8 = 6\] .
 \[ \Rightarrow l = \sqrt {{6^2} + {{4.5}^2}} \]
 \[ \Rightarrow l = \sqrt {36 + 20.25} \]
 \[ \Rightarrow l = 7.5\;m\]
Curved surface area of cone is \[ = \pi rl\]
 \[ = \dfrac{{22}}{7} \times 4.5 \times 7.5\]
 \[ = \dfrac{{742.5}}{7}\]
 \[ = 106.07\;{m^2}\]
Area of canvas= curved surface area of cylinder \[ + \] curved surface area of cone
 \[ = 135.77 + 106.07\]
 \[ = 241.84\;{m^2}\]
Hence, Area of canvas \[ = 241.84\;{m^2}\] .
 \[\therefore \] Option (c) is correct
So, the correct answer is “Option C”.

Note: Before solving this kind of problem, always draw the diagram, it will give you some idea of what needs to be found and values of particular dimensions. Using the above diagram we don’t have the slant height of the cone, but the height of a cone is given. Careful with the substitution. Remember the formulas well.