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Zorluk: OrtaMap Enlargement and Reduction

A wildlife sanctuary occupies an area of 48 cm248\text{ cm}^2 on a topographical map drawn at a scale of 1:100,0001 : 100,000. If the map is reduced to a new scale of 1:400,0001 : 400,000, what is the area of the sanctuary on the new map in cm2\text{cm}^2?

Cevap: 3 cm²

Cevap

The area of the sanctuary on the reduced map is 3 cm23\text{ cm}^2.
When a map scale is reduced from 1:100,0001 : 100,000 to 1:400,0001 : 400,000, the linear dimension reduces by a factor of 44 (since 100,000400,000=14\frac{100,000}{400,000} = \frac{1}{4}). Because area is two-dimensional, the area changes by the square of the linear reduction factor, which is (14)2=116\left(\frac{1}{4}\right)^2 = \frac{1}{16}. Therefore, the area on the new map is 48 cm2×116=3 cm248\text{ cm}^2 \times \frac{1}{16} = 3\text{ cm}^2.

Adım Adım Çözüm

1
Determine the linear scale factor between the original and new scales
Linear scale factor = Original Scale DenominatorNew Scale Denominator=100,000400,000=14\frac{\text{Original Scale Denominator}}{\text{New Scale Denominator}} = \frac{100,000}{400,000} = \frac{1}{4}
When a map scale changes from 1:100,0001 : 100,000 to 1:400,0001 : 400,000, every linear dimension on the map is reduced to 14\frac{1}{4} of its original length.
2
Calculate the area reduction factor
Area scale factor = (14)2=116\left(\frac{1}{4}\right)^2 = \frac{1}{16}
Map area scales as the square of the linear scale factor (n2n^2).
3
Calculate the new area on the reduced map
New Area = 48 cm2×116=3 cm248\text{ cm}^2 \times \frac{1}{16} = 3\text{ cm}^2
Multiplying the original area by the area scale factor gives the area representation on the new map.

Anahtar Kavram

Map Reduction and Area Scale Factor Relationship
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Konular

Map Enlargement and Reduction

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