On a topographical map drawn at a scale of 1 : 100,000, a communications mast is positione… — JAMB UTME

On a topographical map drawn at a scale of $1 : 100,000$ , a communications mast is positioned at point X with an elevation of $180\text{ m}$ , while a valley lookout at point Y has an elevation of $420\text{ m}$ . If the terrain between these two points maintains a constant gradient of $1 \text{ in } 25$ , what is the straight-line distance between point X and point Y on the map?

A
$0.6\text{ cm}$
$6.0\text{ cm}$ Answer
C
$10.5\text{ cm}$
D
$60.0\text{ cm}$

Answer

The distance between point X and point Y on the map is

6.0\text{ cm}

The vertical elevation change between point X (

180\text{ m}

) and point Y (

420\text{ m}

) is

240\text{ m}

. Applying the slope gradient ratio of

1 \text{ in } 25

yields a horizontal ground distance (

HE

) of

240\text{ m} \times 25 = 6,000\text{ m}

(

6\text{ km}

). Given the map scale of

1 : 100,000

1\text{ cm}

on the map represents

100,000\text{ cm}

(

1\text{ km}

) on the ground. Dividing

6\text{ km}

1\text{ km/cm}

gives a map distance of

6.0\text{ cm}

Step-by-Step Solution

Calculate the Vertical Interval (VI) between points X and Y.

VI = 420\text{ m} - 180\text{ m} = 240\text{ m}

The vertical interval represents the elevation difference between the two given points.

Determine the Horizontal Equivalent (HE) on the ground using the gradient ratio.

HE = VI \times 25 = 240\text{ m} \times 25 = 6,000\text{ m} = 6\text{ km}

Gradient is defined as

\text{Gradient} = \frac{\text{Vertical Interval}}{\text{Horizontal Equivalent}}

, so

HE = \text{Vertical Interval} \times \text{Gradient denominator}

Convert the ground distance to map distance using the map scale

1 : 100,000

\text{Map distance} = \frac{6,000\text{ m}}{1,000\text{ m/cm}} = 6.0\text{ cm}

A scale of

1 : 100,000

means

1\text{ cm}

on the map represents

100,000\text{ cm}

(

1,000\text{ m}

1\text{ km}

) on the ground.

Key Concept

Topographic Gradient and Map Scale Distance Calculation

Question

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Step-by-Step Solution

Key Concept

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