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Difficulty: HardContour Lines and Relief Representation

On a topographical map drawn to a scale of 1:25,0001:25,000, a road ascends continuously from Point P at a contour elevation of 180 m180\text{ m} to Point Q at a contour elevation of 430 m430\text{ m}. If the average gradient along this section of the road is 1 in 201\text{ in } 20, what is the distance between Point P and Point Q on the map in centimeters?

  1. A
    2.0 cm2.0\text{ cm}
  2. 20.0 cm20.0\text{ cm}Answer
  3. C
    34.4 cm34.4\text{ cm}
  4. D
    50.0 cm50.0\text{ cm}

Answer

The distance between Point P and Point Q on the map is 20.0 cm20.0\text{ cm}.
The height difference (Vertical Interval) between the two points is 430 m−180 m=250 m430\text{ m} - 180\text{ m} = 250\text{ m}. With a gradient ratio of 1 in 201\text{ in } 20, the horizontal ground distance (Horizontal Equivalent) is 250 m×20=5,000 m250\text{ m} \times 20 = 5,000\text{ m}, which equals 500,000 cm500,000\text{ cm}. Applying the map scale of 1:25,0001:25,000 gives a map measurement of 500,000 cm/25,000=20.0 cm500,000\text{ cm} / 25,000 = 20.0\text{ cm}.

Step-by-Step Solution

1
Calculate the Vertical Interval (VI) between Point P and Point Q
VI=430 m−180 m=250 m\text{VI} = 430\text{ m} - 180\text{ m} = 250\text{ m}
Vertical Interval represents the height difference between the two contour elevations.
2
Calculate the Horizontal Equivalent (HE) on the ground using the given gradient
HE=VI×20=250 m×20=5,000 m\text{HE} = \text{VI} \times 20 = 250\text{ m} \times 20 = 5,000\text{ m}
Gradient is defined as VIHE\frac{\text{VI}}{\text{HE}}. Therefore, HE=VI/Gradient=250 m×20\text{HE} = \text{VI} / \text{Gradient} = 250\text{ m} \times 20.
3
Convert the ground distance (HE) to centimeters
HE in cm=5,000 m×100 cm/m=500,000 cm\text{HE in cm} = 5,000\text{ m} \times 100\text{ cm/m} = 500,000\text{ cm}
Map distance calculations require identical linear units for numerator and denominator.
4
Determine the map distance using the Representative Fraction scale (1:25,0001:25,000)
Map Distance=500,000 cm25,000=20.0 cm\text{Map Distance} = \frac{500,000\text{ cm}}{25,000} = 20.0\text{ cm}
Dividing the actual ground distance by the scale factor gives the equivalent measurement on the map.

Key Concept

Topographic Gradient and Map Scale Conversion
Estimated Time:2m 0s
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Topics

Contour Lines and Relief Representation

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