GATE ECE 2015 Set 3 | GA | Question: 8

Question

From a circular sheet of paper of radius $30$ cm, a sector of $10\%$ area is removed. If the remaining part is used to make a conical surface, then the ratio of the radius and height of the cone is ________.

Answer 1

Let

• radius of circular sheet of paper $= R$
• radius of the cone $=r$
• height of cone $= H$

Perimeter of base of cone $= 0.9\times 2\pi R$
$\implies 2\pi r  = 0.9*2\pi R$
$\implies r  = 0.9R$

Now, height of cone $H  = \sqrt{R^{2}-r^{2}}$
$\implies H  = r.\sqrt{(R/r)^{2}-1}$
$\implies r/H= \frac{1}{\sqrt{(1/0.9)^{2}-1}}$
$= 2.06$

Comments

how perimeter of the cone will be equal to the 90% area of the circle ? perimeter has some unit say meter whereas area has unit meter$^2$ then how both can be equal.

If you are not getting it then solve it by taking area instead of perimeter.

Since, $90\%$ area of the circular sheet is used to make canonical surface.

So,  $90\%$ area of the circular sheet = curved surface area of cone.

   $0.90 \times \pi R^{2} = \pi r l$

Now, here slant height $l$ will be same as radius of the circular sheet i.e. $R=30 \;cm$

So, $0.90 \times \pi R^{2} = \pi r R$

So, $r = 27$

Now, $h= \sqrt{l^{2}-r^{2}} = \sqrt{30^{2}-27^{2}} = 13.08$

So, $\frac{r}{h} = \frac{27}{13.08}$ = $2.06$

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Derivation of Surface Area of a cone(YouTube video)

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sector to cone:-->
https://youtu.be/Z5cbInWfWoA?si=9K5gfoJ23nZCWCsC