GATE2017 ME-1: GA-3

Question

A right-angled cone (with base radius $5$ cm and height $12$ cm), as shown in the figure below, is rolled on the ground keeping the point $P$ fixed until the point $Q$ (at the base of the cone, as shown) touches the ground again.

By what angle (in radians) about $P$ does the cone travel?

• A. $\frac{5\pi}{12}$
• B. $\frac{5\pi}{24}$
• C. $\frac{24\pi}{5}$
• D. $\frac{10\pi}{13}$

Comments

How you got angle 360 for 2pi * 13??

---

option d.

it will sweep horizontal distance equal to circumfrence=2*pi*r=2*pi*5=10pi

now, distance of point Q from P(using pythagoras)=sqrt(12^2 + 5^2)=13..

now during it course of journey, it will form circular arc, where

'P' being center of arc, '10pi' being length of arc and '13' being radius of circular arc..

now using formula for circular arc i.e

theta = l/r

theta = 10pi/13...

i will suggest you first draw a proper diagram, then you will get exactly what i have done..

---

The original question was:

A right-angled cone (with base radius 55 cm and height 1212 cm), as shown in the figure below, is rolled on the ground keeping the point PP fixed until the point QQ (at the base of the cone, as shown) touches the ground again.

By what angle (in radians) about PP does the cone travel?

 

So you're saying that this is same as opening the cone?

Answer 1

Point $Q$ will touch again, when cone will roll around $P$ and will travel arc length of $2\pi r$

So, arc length made by cone $= 2 \pi r = 10\pi $ cm

If the cone rotates one round around point $P$ it will cover perimeter of length $2 \pi l$ cm

where, $l = \sqrt{h^2+r^2} = 13$ cm

So, perimeter of one rotation $= 26 \pi$ cm

Thus, the angle(in radian) which cone makes $= \frac{10\pi}{26\pi}\times 360\times \frac{\pi}{180} = \frac{10\pi}{13}$

Correct Answer: $D$

Comments

@Naveen Kumar 3

Explain this part

 

Thus, the angle(in radian) which cone makes =10π/26π×360×π/180=10π/13

---

@jlimbasiya, when cone will rotate one round( $2\pi l=26\pi$ distance) around $P$, then it will make $360^\circ$.Here, when point $Q$ again touches ground, it covers $10\pi$ arc length.

So, angle made by it = $\frac{10\pi}{26\pi}*360^\circ$   (this is in degree)

so, convert this result by multiplying with $\frac{\pi}{180}$ to convert it in radians.

---

We can also directly multiply by 2π. Last second line in best answer

= (10π / 26 ) * 2 π

= 10 π / 13

So no confusion of degree and radian conversion and all that.

Answer 2

Base radius = 5cm, Height = 12m. So Slant Height = $\sqrt{5^{2}+12^{2}}$ = 13cm.

We open it now. I appears as an arc if opened. Radius of the arc = Slant Height of the cone = 13cm.

Length of the arc = Circumfernce of the cone = 2*π*5 = 10πcm

Now, when, length of arc is 2*π*13, angle subtended is 2π.

So, angle subtended when length of arc is 10π = $\frac{2π}{26π} * 10π$ = $\frac{10π}{13}$