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Zorluk: ZorMap Enlargement and Reduction

On Map X, drawn at a scale of 1:25,0001 : 25,000, a planned agricultural settlement occupies a square grid block measuring 4 cm4\text{ cm} by 4 cm4\text{ cm}. Map X is reduced to produce Map Y such that the same agricultural settlement occupies a reduced area of 1 cm21\text{ cm}^2. What is the Representative Fraction (R.F.) scale of Map Y?

  1. A
    1:6,2501 : 6,250
  2. B
    1:50,0001 : 50,000
  3. 1:100,0001 : 100,000Cevap
  4. D
    1:400,0001 : 400,000

Cevap

The Representative Fraction scale of Map Y is 1:100,0001 : 100,000.
The original grid block has an area of 16 cm216\text{ cm}^2 on Map X. Since the reduced area on Map Y is 1 cm21\text{ cm}^2, the area reduction factor is 1616. The linear reduction factor is the square root of the area factor, which is 16=4\sqrt{16} = 4. Reducing linear dimensions by a factor of 44 makes the map scale smaller by a factor of 44, increasing the scale denominator from 25,00025,000 to 100,000100,000. Thus, Map Y has a scale of 1:100,0001 : 100,000.

Adım Adım Çözüm

1
Calculate the original area of the settlement on Map X.
Area on Map X=4 cm×4 cm=16 cm2\text{Area on Map X} = 4\text{ cm} \times 4\text{ cm} = 16\text{ cm}^2.
Map area for a rectangular or square feature is length multiplied by width.
2
Determine the area scale change ratio.
\text{Area Ratio} = \frac{\text{Area on Map Y}}{\text{Area on Map X}} = \frac{1\text{ cm}^2}{16\text{ cm}^2} = \frac{1}{16}.
Area scale ratio is the ratio of final map area to initial map area.
3
Determine the linear scale reduction factor.
\text{Linear Factor} = \sqrt{\frac{1}{16}} = \frac{1}{4}.
Linear scale factor is the square root of the area scale factor (n=n2n = \sqrt{n^2}).
4
Calculate the new Representative Fraction scale denominator for Map Y.
\text{New Scale Denominator} = 25,000 \times 4 = 100,000 \Rightarrow \text{Scale of Map Y} = 1 : 100,000.
When a map is reduced, its scale denominator increases proportionally by the reciprocal of the linear scale factor.

Anahtar Kavram

Relationship between linear scale factor and area scale factor in map reduction
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Konular

Map Enlargement and Reduction

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