Question

Their are two circles with Center A and B with radius
\(40\) m and \(80\) m respectively such that they meet at point x externally.
Bolt runs at \(80\pi\) m/min on circle A.
Paul runs at \(40\pi\) m/min on circle B.
Both of them start at point x and run multiple rounds.
If Bolt gives Paul start of \(4\) min exactly before he starts running himself.
After how much time will the straight line between them will be \(240\) m.

Answer 1

that looks hard
http://i.ytimg.com/vi/ItWI9nVtS-s/maxresdefault.jpg

Answer 2

no

Answer 3

So, the point where two points meet is known as the vertex, yes?

Answer 4

I get 1 min

Answer 5

so just one question i have, at which direction P and B runs ?
like we have 4 choices here i can't determine which direction to take :P

Answer 6

I don't think direction matters in this case

Answer 7

direction doesnt matter

Answer 8

well :P
lets solve it geometrically

(i don't know yet if its matter or not at least it matter to me )

Answer 9

\(\large \color{black}{ a.)\quad 6 \quad min\hspace{.33em}\\~\\
b.)\quad 10 \quad min\hspace{.33em}\\~\\
c.)\quad 12.5 \quad min\hspace{.33em}\\~\\
d.)\quad none \quad \hspace{.33em}\\~\\
}\)

Answer 10

so as been said 240 m
as graph this is the only case that could be (y,z)
it cant be x1 and z1(at any place since distance would be less than 240)

Answer 11

That's right. And it take s them both 1 minute to get to y and z respectively

Answer 12

:)

Answer 13

so i can say only in this case direction seems does not matter otherwise ;P it is

Answer 14

@alekos if u think its \(1\) min , then prove it.

Answer 15

The two circles are tangent?

Answer 16

ok so paul start at x, parameter of the circle B \(160\pi\)
so for full loop
time = 160 pi/ 40 pi = 4 mints

for Bolt start at x , parameter of the circle A \(80 \pi\)
so for full loop
time=80 pi/80 pi=1 mint

Answer 17

Theres your proof!

Answer 18

so at 4 mints both start from x
and they can reach y,z at these time (after 4 mints )
Pual Bolt
2 0.5
6 1.5
10 2.5
... ...

hmmm idk if they can ever met :O

Answer 19

the earlier work done by ikram shows \(1\) min for bolt to reach at point x.

Answer 20

So they have got to be diametrically opposite to do so. We can use the equations of circular motion, I think...\[\theta = \omega t \]Here, \(\omega_{Bolt } = 2\pi ~\rm rad/min\) and \(\omega_{Paul} = \pi/4 ~\rm rad/min\).
So the number of minutes will have to be a decimal.

Answer 21

Why decimal?

If it is integral, then Bolt will be at point x.

Answer 22

yeah @mathmath333 so i would say they never meet :P
i'll see what mira and pk have :)