Question

Find the length of cloth used in making a conical pandal of height 100m and base radius 240m, if the cloth is $100\pi $m wide.

Answer 1

Hint: Visualise the structure of the cone and break it down to basic structures. Then calculate the area . A cone is a three dimensional structure that has a circle as its base.

Complete step by step answer:
The surface area of a cone is the area of its circular base plus the area of its curved surface.
Radius of the circular base is r.
Area of the circular base is $\pi {{r}^{2}}$
Height of the cone is h and l is known as the slant height of the cone.
Area of the curved surface of the cone is $\pi rl$
Where $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$(This is because the triangle formed by r,h and l is a right angled triangle and l is the hypotenuse )
Total surface area of cone= Area of circular base +Area of curved surface
Total surface area of cone= $\pi {{r}^{2}}$+ $\pi rl$=$\pi r\left( r+l \right)$
Now substituting $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$, we get
Total surface area of cone=$\pi r\left( r+\sqrt{{{r}^{2}}+{{h}^{2}}} \right)$
Where r is the radius of the circular base of the cone and h is the height of the cone.

According to the question, h=100m and r=240m
Now we know that $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$
$l=\sqrt{{{240}^{2}}+{{100}^{2}}}$
$l=\sqrt{67600}$
$l=260$m
As we know that, the curved surface area of a cone=$\pi rl$
=$\pi \times 240\times 260$
$=62400\pi $${{m}^{2}}$
The curved surface area of the conical pandal is equal to the area of the cloth used for conical pandal.
Area of cloth $=62400\pi $${{m}^{2}}$=length of cloth×width of cloth
Length of cloth=area of cloth/width of cloth
Length=$\dfrac{62400\pi }{100\pi }$= 624 m
Hence the length of the cloth is 624 m.

Note: Keep in mind the units of the dimensions. Most of the mistakes are made with units.
Also time can be saved if the formula of total surface area and curved surface area of cone is remembered.