Question

What length of tarpaulin 3m wide will be required to make a conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm.

Answer 1

Hint:The tent is in conical shape and Tarpaulin will be in the shape of a rectangle. So the tarpaulin will be on the curved surface of the tent. Find the slant height of the tent using its height and radius and then find the curved surface area of the tent. Tarpaulin material will be in rectangle shape. Find the length of the tarpaulin material by equating the area of the tarpaulin with the curved surface area of the conical tent. Curved surface area of the tent must be equal to the area of the tarpaulin.
Formulas used: Curved surface area of a cone is $\pi rL$ where r is the base radius and L is the slant height.
Area of a rectangle is $l \times b$ where l is the length and b is the breadth.

Complete answer:
We are given that the radius of the tent is 6m and height is 8m and width of the tarpaulin is 3m.
r=6m, h=8m
As we can see, the slant height of the tent is not given so we need to find the slant height of the tent.
When we join the vertex of the cone perpendicularly with its base, then we will get a right angled triangle.In a right angled triangle, Pythagoras theorem is used to find the values of the sides. So, that is why we have to use Pythagoras theorem, which is that the Hypotenuse square is equal to the sum of the squares of the remaining two adjacent sides, to get the slant height.
Here hypotenuse is the slant height and adjacent sides are radius and height.
${L^2} = {h^2} + {r^2}$ , where L is the slant height
$
  {L^2} = {8^2} + {6^2} \\
  {L^2} = 64 + 36 \\
  {L^2} = 100 \\
  L = \sqrt {100} = 10m \\
 $
Therefore, slant height L=10m
Curved surface area of the conical tent = $\pi rL$
$
   = \pi \times r \times L \\
   = \dfrac{{22}}{7} \times 6 \times 10 \\
   = \dfrac{{22 \times 60}}{7} \\
   = \dfrac{{1320}}{7}{m^2} \\
 $
Area of the tarpaulin material = Area of the conical tent
Length × breadth= Area of the conical tent
Given width (breadth) = 3m
$
  l \times 3 = \dfrac{{1320}}{7} \\
  l = \dfrac{{1320}}{{3 \times 7}} \\
  l = \dfrac{{1320}}{{21}} \\
  l = 62.8m \\
 $
Therefore, the length of tarpaulin material obtained is 62.8m.
The extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm.
Final length of the tarpaulin material after adding the extra length for margins and wastage is
$
   = 62.8m + 20cm \\
  1m = 100cm \\
  1cm = \dfrac{1}{{100}}m \\
  20cm = \dfrac{{20}}{{100}}m \\
   = 62.8m + \dfrac{{20}}{{100}}m \\
   = \left( {62.8 + 0.2} \right)m \\
   = 63m \\
 $
Therefore, the required length of tarpaulin 3m wide to make a conical tent of height 8m and base radius 6m is 63m.

Note:Curved surface area of a cone and total surface area of a cone are different. Curved surface area of a cone is $\pi rL$ whereas Total surface area of a cone is $\pi rL + \pi {r^2}$ .