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Difficulty: MediumMap Scale Conversion and Distance Calculation

A forest reserve occupies an area of 5 cm25\text{ cm}^2 on a topographical map drawn to a scale of 1:100,0001 : 100,000. If the map is enlarged to a new scale of 1:50,0001 : 50,000, what is the area of the forest reserve on the new map?

  1. A
    10 cm210\text{ cm}^2
  2. 20 cm220\text{ cm}^2Answer
  3. C
    2.5 cm22.5\text{ cm}^2
  4. D
    1.25 cm21.25\text{ cm}^2

Answer

The area of the forest reserve on the new enlarged map is 20 cm220\text{ cm}^2.
When a map scale is changed from 1:100,0001 : 100,000 to 1:50,0001 : 50,000, the linear dimensions are enlarged by a factor of 100,00050,000=2\frac{100,000}{50,000} = 2. Because area is measured in two dimensions, the area scale factor is the square of the linear scale factor (22=42^2 = 4). Therefore, the new area on the map is 5 cm2×4=20 cm25\text{ cm}^2 \times 4 = 20\text{ cm}^2.

Step-by-Step Solution

1
Determine the linear scale enlargement factor
Linear Factor = Old Scale DenominatorNew Scale Denominator=100,00050,000=2\frac{\text{Old Scale Denominator}}{\text{New Scale Denominator}} = \frac{100,000}{50,000} = 2
Enlarging from 1:100,0001 : 100,000 to 1:50,0001 : 50,000 doubles the linear dimensions on the map.
2
Calculate the area scale factor
Area Factor = (Linear Factor)2=22=4(\text{Linear Factor})^2 = 2^2 = 4
Areal change on a map is proportional to the square of the linear scale change.
3
Compute the new area on the enlarged map
New Area = Original Area×Area Factor=5 cm2×4=20 cm2\text{Original Area} \times \text{Area Factor} = 5\text{ cm}^2 \times 4 = 20\text{ cm}^2
Multiplying the initial map area by the area scale factor yields the new map area.

Key Concept

Relationship between Linear Scale and Area Scale in Map Enlargement
Estimated Time:1m 30s
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Topics

Map Scale Conversion and Distance Calculation

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