Examkin

Question

Difficulty: MediumMap Enlargement and Reduction

A wildlife sanctuary covers an area of 36 cm236\text{ cm}^2 on Map X, which is drawn to a scale of 1:40,0001 : 40,000. If the map is reduced to a scale of 1:120,0001 : 120,000, what is the area of the sanctuary on the new map?

  1. A
    12 cm212\text{ cm}^2
  2. 4 cm24\text{ cm}^2Answer
  3. C
    108 cm2108\text{ cm}^2
  4. D
    324 cm2324\text{ cm}^2

Answer

The area of the sanctuary on the new map is 4 cm24\text{ cm}^2.
When a map is reduced from a scale of 1:40,0001 : 40,000 to 1:120,0001 : 120,000, the linear dimensions become 40,000120,000=13\frac{40,000}{120,000} = \frac{1}{3} of the original size. Because area is proportional to the square of linear dimensions, the area factor is (13)2=19\left(\frac{1}{3}\right)^2 = \frac{1}{9}. Reducing 36 cm236\text{ cm}^2 by a factor of 9 gives 4 cm24\text{ cm}^2.

Step-by-Step Solution

1
Determine the linear scale factor of reduction
Linear scale factor k=Old Scale DenominatorNew Scale Denominator=40,000120,000=13k = \frac{\text{Old Scale Denominator}}{\text{New Scale Denominator}} = \frac{40,000}{120,000} = \frac{1}{3}
Going from 1:40,0001 : 40,000 to 1:120,0001 : 120,000 reduces all linear map dimensions to one-third of their original length.
2
Calculate the area scale factor
Area scale factor k2=(13)2=19k^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}
Surface area changes proportionally to the square of the linear scale factor.
3
Calculate the new map area
New Area = Original Area ×k2=36 cm2×19=4 cm2\times k^2 = 36\text{ cm}^2 \times \frac{1}{9} = 4\text{ cm}^2
Multiplying the original map area by the area scale factor gives the area on the reduced map.

Key Concept

Map Reduction and Area Scale Relationship
Estimated Time:1m 30s
Rate this question

Topics

Map Enlargement and Reduction

More questions from this topic