Variables:
- L = length of the train in meters
- v = speed of the train in meters per second (normal speed)
- t1 = time taken by the train to pass the 206-meter long bridge at normal speed
- t2 = time taken by the train to pass the 170-meter long bridge at one-third of the normal speed
From the problem statement, we have two equations involving these variables:
Equation 1: L + 206 = v * t1
Equation 2: L + 170 = (1/3)*v * t2
We can solve this system of equations for L by eliminating v. To do this, we can rearrange Equation 1 to solve for v:
v = (L + 206) / t1
Then, we can substitute this expression for v into Equation 2:
L + 170 = (1/3)*[(L + 206) / t1] * t2
Multiplying both sides by 3*t1 and simplifying, we get:
3*L*t1 + 3*170*t1 = (L + 206)*t2
Expanding the right-hand side and simplifying, we get:
3*L*t1 + 510 = L*t2 + 206*t2
Now, we can solve for L by rearranging terms and simplifying:
L = (206*t2 - 3*170*t1) / (3*t1 - t2)
Substituting the given values, we get:
L = (206*45 - 3*170*17) / (3*17 - 45) = 510
The length of the train is 510 meters.