A railroad track is laid along side the arc of a circle of radius 1800 ft. The circular part of the track subtends a central angle of 40 degrees. How long (in seconds) will it take a point on the front of a train traveling 30 mph to go around this portion of the track
To find the time it takes for a point on the front of the train to go around this portion of the track, we need to calculate the arc length. The formula for the arc length is:
Arc Length = radius * central angle
Given that the radius is 1800 ft and the central angle is 40 degrees:
Arc Length = 1800 ft * 40 degrees
However, the formula for arc length assumes that the angle is in radians, not degrees. To convert degrees to radians, we need to multiply by π/180.
Arc Length = 1800 ft * (40 degrees * π/180)
Simplifying the expression:
Arc Length = 1800 ft * (40 * π/180)
Now, the next step is to calculate the time it takes for the train to cover this distance. We know the speed of the train is 30 mph, which stands for miles per hour. However, the arc length is given in feet, so we need to convert the speed to feet per second.
We can convert miles to feet by multiplying by 5280 (since there are 5280 feet in a mile), and we can convert hours to seconds by multiplying by 3600 (since there are 3600 seconds in an hour).
Speed = 30 mph * (5280 ft/mile) / (3600 s/hour)
Simplifying the expression:
Speed = 44 ft/s
Now we can find the time it takes for the train by dividing the arc length by the speed:
Time = Arc Length / Speed
Plugging in the values:
Time = (1800 ft * (40 * π/180)) / 44 ft/s
Simplifying the expression:
Time = (1800 * π * 4/9) / 44 s
Now, we can calculate this value:
Time ≈ 58.19 seconds
Therefore, it will take approximately 58.19 seconds for a point on the front of a train traveling at 30 mph to go around this portion of the track.
the length of the arc is ... (40 / 360) * 2 * π * 1800
30 mph is 44 feet per second