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Difficulty: HardMap Scale Conversion and Distance Calculation

A rectangular agricultural zone measures 6 cm6\text{ cm} by 4 cm4\text{ cm} on a topographical map drawn to a Representative Fraction (R.F.) scale of 1:100,0001 : 100,000. If the map is enlarged so that the area of the agricultural zone on the new map becomes 96 cm296\text{ cm}^2, what is the Representative Fraction (R.F.) scale of the enlarged map?

  1. 1:50,0001 : 50,000Answer
  2. B
    1:25,0001 : 25,000
  3. C
    1:200,0001 : 200,000
  4. D
    1:400,0001 : 400,000

Answer

The Representative Fraction (R.F.) scale of the enlarged map is 1:50,0001 : 50,000.
The correct answer of 1:50,0001 : 50,000 is derived by recognizing that the original map area of 24 cm224\text{ cm}^2 increases to 96 cm296\text{ cm}^2, representing a four-fold areal enlargement. Because linear dimensions change as the square root of the areal change, the linear scale factor is 4=2\sqrt{4} = 2. Enlarging a map by a factor of 2 doubles linear dimensions on paper, making the scale larger by reducing the scale denominator from 100,000 to 50,000.

Step-by-Step Solution

1
Calculate the area of the agricultural zone on the original map.
Original Area =6 cm×4 cm=24 cm2= 6\text{ cm} \times 4\text{ cm} = 24\text{ cm}^2.
Knowing the original map area allows us to determine the area enlargement ratio.
2
Determine the area scale change ratio.
Area Ratio =New AreaOriginal Area=96 cm224 cm2=4= \frac{\text{New Area}}{\text{Original Area}} = \frac{96\text{ cm}^2}{24\text{ cm}^2} = 4.
The area of the zone has been enlarged 4 times.
3
Calculate the linear scale enlargement factor.
Linear Factor =Area Ratio=4=2= \sqrt{\text{Area Ratio}} = \sqrt{4} = 2.
Linear scale changes as the square root of areal scale changes.
4
Calculate the scale denominator for the enlarged map.
New Scale Denominator =Original Scale DenominatorLinear Factor=100,0002=50,000= \frac{\text{Original Scale Denominator}}{\text{Linear Factor}} = \frac{100,000}{2} = 50,000.
Enlarging a map increases its detail and linear dimensions, which reduces the scale denominator proportionally.

Key Concept

Map Enlargement and Linear vs Areal Scale Relationship
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Topics

Map Scale Conversion and Distance Calculation

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