Travelling Ant

Geometry Level 2

An ant finds itself trapped in the $xy$ -plane, and its initial position is $(1,0).$

Let $S_k$ denote the circle with radius $k$ centered around the origin. Starting from $(1,0)$ , the ant walks 1 unit counter-clockwise on $S_{1}.$ Then, it walks directly (radially outward) to $S_2,$ on which it will walk 2 units counter-clockwise. Then, it will walk directly to $S_3$ and walk 3 units counter-clockwise, and so, with the ant walking $k$ units on $S_k.$ (See the image above.)

When the ant crosses the positive $x$ -axis for the first time since it left $(1,0)$ , it is on $S_{n}$ . What is $n$ ?

The answer is 7.

8 solutions

Test User
Feb 20, 2014

Angular distance is equivalent to $\frac{arc length}{radius}$ and therefore the ant is always traveling $\frac{1}{1} = \frac{2}{2} = \frac{3}{3} = \ ... \ = 1 \ radian$ per circle. There are $6.28 \ radians$ in a circle, therefore while on circle number $\boxed{7}$ the ant will cross the positive x-axis.

This is a IITJEE question

Tejas Rangnekar - 7 years, 3 months ago

Which year's IIT-JEE paper ?

Vishal Yadav - 5 years, 8 months ago

when the ant crosses to the fourth quadrant, i think he is already in the positive direction of the x- axis, so my answer is 5, or maybe my assumption is wrong........

Alan Ramos - 7 years, 3 months ago

The question says, 'When the ant crosses the positive x-axis...', not 'when the ant is along the positive x-axis'.

I hope this solves your doubt. :)

Aniruddha Bhattacharjee - 5 years, 2 months ago

brilliant explanation !!!

p s - 7 years, 3 months ago

I used the same solution to solve this problem Nice!

Rindell Mabunga - 7 years, 3 months ago

couldn't understand the question at first. damn myself, so easy it was.

Anirban Ghosh - 7 years, 3 months ago

thanks for the explanation :D very well explained~

Heng Joe Kit - 7 years, 3 months ago

in the question,u mentioned 1 unit, not 1 radian......so if we will take 1 unit =1/2 radian the ans will differ......so it is an incomplete question.

Ashish Kumar - 7 years, 3 months ago

Can you be more clear? I mentioned 1 unit, because a unit distance is travelled. Also, how is 1 unit =1/2 radian?

Vaibhav Nayak - 7 years, 3 months ago

no ,i m not telling that 1 unit =1/2radian.....but it can be ,because u didn't mention any thing about 1 unit, in the circle......see u have taken 1 unit = 1radian!! why??

Ashish Kumar - 7 years, 3 months ago

@Ashish Kumar – Because for $S_n$ , the ant travel $n$ unit along arc. Radius of $S_n$ and distance traveled is always same. Knowing that $radius\cdot\theta = arc\:length\\\theta = \dfrac{radius}{arc\:length} = 1\:radian$

Micah Wood - 7 years, 3 months ago

Kishore Reddy
Feb 28, 2014

The ant covers k units on Sk circle,and the radius of this circle is k units

so angle covered by ant on every circle is

k * angle=k (radius * angle=length of arc)

angle=1 radian =57.29 degree

for crossing +ve X-axis the ant must cover 360 degree

so n * 57.29>360

n=7(since n is an integer)

নিটুল হালদার
Mar 6, 2014

every time it travels 1 radian with respect to origin. So it needs 2 PI radin to be on the positive X direction which needs integer more than 6.28(2 PI) that is 7.

Harendra Pal Singh
Mar 2, 2014

total angular distance to be travelled is 2 pi radian. ant travels one radian on each circle, thus will travel 6 radian on sixth circle and will cross positive direction of x axis on seventh circle.

Rishav Koirala
Mar 1, 2014

Travelling on each circle, the ant covers one radian of angular distance. To take one complete round and cross the positive direction of the x axis, the ant will complete 2π radians. 1st circle----------> 1 radian or 1 radian ----------> 1st circle So, 2π radian = 2 x 3.14 = 6.28 So the ant will be travelling on the 7th circle

Shristi Kumari
Mar 3, 2014

the ant has to cover the arc > 6 units(-3 to +3)to cross the +x axis.thus,on it covers 7 units on 7th circle.

Siva Senthil
Mar 1, 2014

just make a table of travelled length from starting point and circumference of consecutive circles. you will find the answer at when the travelled length becomes more than the circumference of the respective circle.

Som Debasis
Feb 27, 2014

For each circle,

the ant covers (r /2 * pi * r) * 2 * pi radians

i.e. 1 radian.

So to cover 2*pi(=6.28) radian he will do it in 7th circle.

6 radians in 6 circles

6.28 will be covered in 7th