Question

What length of canvas \[3m\] wide will be required to make a conical tent of height \[8m\] and radius of base \[6m\]. (Use\[\pi = 3.14\])

Answer 1

Hint:
Here we will use the formula of the curved surface area of the cone to find out the length of the canvas. Firstly we will find out the slant height of the cone. Then we will find out the surface area of the cone. Then we will equate this surface area of the cone with the area of the canvas by which we will get the length of the canvas.

Formula used: Curved surface area of the cone \[ = \pi rl\] where, \[r\] is the radius of the cone, \[l\] is the slant height of the cone.

Complete step by step solution:
It is given that the height of the cone is \[8m\] and radius of the base of the cone is \[6m\].
Firstly we will find out the slant height of the cone. Therefore
\[slant\,height = \sqrt {{{\left( {height} \right)}^2} + {{\left( {radius} \right)}^2}} \]
\[l = \sqrt {{8^2} + {6^2}} = \sqrt {64 + 36} = \sqrt {100} = 10m\]
Now we will find the curved surface of the cone by putting the value of slant height and radius in the formula, we get
Curved surface area of the cone \[ = \pi rl = 3.14 \times 6 \times 10 = 188.4{m^2}\]
Now we know that the area of the canvas must be equal to the curved surface area of the cone. Therefore, we get
\[Area\,of\,canvas = Curved\,surface\,area\,of\,cone\]
\[ \Rightarrow L \times B = \pi rl\]
It is given that the canvas is \[3m\] wide i.e. \[B = 3m\]. Therefore, by putting the values in this we get
\[ \Rightarrow L \times 3 = 188.4\]
\[ \Rightarrow L = \dfrac{{188.4}}{3} = 62.8m\]

Hence, the length of the canvas is \[62.8m\]

Note:
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the vertex. Surface area is the sum of all the areas of the faces of an object or shape. Volume is the amount of space occupied by an object in three-dimensional space. Volume is generally measured in cubic units.
Total surface area of the cone \[ = \pi rl + \pi {r^2}\] where \[r\] is the radius of the cone,\[l\] is the slant height of the cone and \[h\] is the height of the cone.
Volume of a cone \[ = \dfrac{{\pi {r^2}h}}{3}\] where,\[r\] is the radius of the base of the cone and \[h\] is the height of the cone.