Get On Track -- Math Brainteaser (Suitable for Experienced Algebra Students)

[Otis]

Runners Laura and Betty begin at diametrically opposite points on a circular track and run in opposite directions around the track. Each runner runs at a constant speed. Laura and Betty meet for the first time after Laura has run 170 meters. They meet for a second time after Betty has run 90 meters past their first meeting point. How many meters long is the track?

Spoiler: Hints

I focused on the semicircle in which they both began running. I followed how far Betty ran, to reach each meeting point. Three arclengths make up the semicircle, and we may write expressions for each of them.

?

[imath]\;[/imath]

 

[Harry_the_cat]

Spoiler

Arc length of semicircle \(\displaystyle > 170\)m.
Let \(\displaystyle x\) be arc length of semicircle.
Let \(\displaystyle v_L\) and \(\displaystyle v_B\) be Lauren's and Betty's (constant) velocity respectively.
\(\displaystyle s =v*t\).

For first time interval \(\displaystyle t_1\):
\(\displaystyle 170 = v_L*t_1\) and \(\displaystyle x - 170 = v_B*t_1\)
\(\displaystyle \frac{v_L}{v_B} = \frac{170}{x - 170}\)

For second time interval \(\displaystyle t_2\):
\(\displaystyle 90 = v_B*t_2\) and \(\displaystyle 2x - 90 = v_L*t_2\)
\(\displaystyle \frac{v_L}{v_B} = \frac{2x - 90}{90}\)

Since both velocities are constant, equating these gives:

\(\displaystyle \frac{170}{x - 170} = \frac {2x -90}{90}\)
\(\displaystyle (2x - 90)(x - 170) = 170*90\)
\(\displaystyle 2x^2-430x+170*90 = 170*90\)
\(\displaystyle 2x(x - 215)=0\)
\(\displaystyle x=215\) since \(\displaystyle x>0\)

The the length of the track is \(\displaystyle 2*215 = 430\) metres.

 

[lev888]

Spoiler

Their first segments covered half the track, second - the whole track. This means each covered twice the distance in the second segments.
Laura's second segment is 170*2=340m. Total track length: 340+90 = 430m.

 

[Harry_the_cat]

Spoiler

Their first segments covered half the track, second - the whole track. This means each covered twice the distance in the second segments.
Laura's second segment is 170*2=340m. Total track length: 340+90 = 430m.

Well that's an easier method than mine!! ?

 

[Otis]

that's an easier method than mine!

You both did good!

Your approach matches what I'd first starting doing, but then I had a lucky realization. (For all I know, most students reading the spoilers would understand your logic before the others.)

I think I came to realize what Lev did, albeit slightly differently (leading me to do a bit more work than he did, heh).

Spoiler

P = semicircle arc length

Betty ran P-170 meters, to the first meeting point.

Once I realized that Laura would need to run two more "segments", to meet Betty a second time, I knew that Betty's 90-meter segment was 2(P-170). Hence,

P = 3(P - 170) + 80

[imath]\;[/imath]