A proposed high-voltage transmission line connecting a hydro-electric station to an urban … — JAMB UTME

A proposed high-voltage transmission line connecting a hydro-electric station to an urban sub-station measures $14.5\text{ cm}$ on a topographical map drawn to a scale of $1 : 50,000$ . If the map undergoes a photographic reduction such that its total surface area becomes $\frac{1}{4}$ of its original size, what is the actual ground distance, in kilometers, represented by a length of $5.8\text{ cm}$ measured on the newly reduced map?

Answer: 5.8 km

Answer

The actual ground distance represented by 5.8 cm on the reduced map is 5.8 kilometers.

Reducing the surface area of a map to one-fourth reduces its linear dimensions to one-half, because linear scale factor equals the square root of the area scale factor. With linear dimensions halved, each centimeter on the new map represents twice as much ground distance as before, changing the scale from 1 : 50,000 to 1 : 100,000. At a scale of 1 : 100,000, 1 cm represents 1 km. Therefore, 5.8 cm on the reduced map corresponds to an actual ground distance of 5.8 km.

Step-by-Step Solution

Determine the linear scale factor from the area reduction ratio

Linear factor = \sqrt{\frac{1}{4}} = \frac{1}{2}

Map area varies as the square of the linear scale factor. Reducing area to one-fourth reduces linear dimensions by half.

Calculate the Representative Fraction (R.F.) of the reduced map

New R.F. = 1 : (50,000 \times 2) = 1 : 100,000

Reducing the physical dimensions of a map by half doubles the denominator of its scale ratio, making it a smaller scale map.

Convert the new R.F. scale to kilometers per centimeter

1\text{ cm} = 100,000\text{ cm} = 1.0\text{ km}

Dividing the scale denominator (100,000) by 100,000 converts centimeters to kilometers.

Compute the ground distance for a map length of 5.8 cm

5.8\text{ cm} \times 1.0\text{ km/cm} = 5.8\text{ km}

Multiplying map distance by the ground equivalence per centimeter yields actual distance.

Key Concept

Linear vs Area Scale Conversion and Ground Distance Calculation

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