If the sleeper density is M + 7 on a broad-gauge route, what is the number of sleepers required for 1.3 km length? (where M is the length of the rail in metres)
2000
Sleeper Calculation for Broad-Gauge Railway Tracks
This solution provides a detailed step-by-step calculation to determine the total number of sleepers required for a specific length of a broad-gauge route, considering a given sleeper density formula. Understanding the standard length of a rail and the concept of sleeper density is fundamental for accurate railway track infrastructure planning.
Rail Length for Broad-Gauge Tracks (M)
The question mentions 'M' as the length of the rail in meters. For a broad-gauge route, especially in Indian railways, the standard length of a rail section is conventionally considered to be 13 meters. This value of 'M' is a crucial input for further calculations related to sleeper density.
- Standard rail length (\(M\)) for broad gauge = \(13\) meters
Sleeper Density Determination
The problem provides a formula for the sleeper density as \(M + 7\). This formula indicates the number of sleepers laid per standard rail length. By substituting the standard rail length 'M' into this formula, we can find the exact sleeper density.
- Given sleeper density formula = \(M + 7\)
- Substitute the value of \(M = 13\):
- Sleeper density = \(13 + 7 = 20\) sleepers per rail length
Route Length Conversion to Meters
The total length of the broad-gauge route is given in kilometers. To ensure consistency in units with the rail length (which is in meters), it is necessary to convert the total route length from kilometers to meters.
- Given total length of the broad-gauge route = \(1.3\) km
- Conversion factor: \(1 \text{ km} = 1000 \text{ meters}\)
- Total length in meters = \(1.3 \times 1000 = 1300\) meters
Number of Rail Lengths in the Route
To find out how many standard rail lengths are present in the total specified route length, we divide the total route length in meters by the length of a single standard rail.
- Number of rail lengths = \(\frac{\text{Total length in meters}}{\text{Length of one rail (M)}}\)
- Number of rail lengths = \(\frac{1300 \text{ meters}}{13 \text{ meters/rail length}} = 100\) rail lengths
Total Number of Sleepers Required
Finally, to calculate the total number of sleepers needed for the entire 1.3 km broad-gauge route, we multiply the number of standard rail lengths by the previously calculated sleeper density.
- Total number of sleepers = Number of rail lengths \(\times\) Sleeper density
- Total number of sleepers = \(100 \times 20 = 2000\)
Therefore, the total number of sleepers required for the 1.3 km broad-gauge route with a sleeper density of \(M + 7\) is 2000.
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