A rectangular agricultural estate measuring 5\text{ cm} by 12\text{ cm} on a topographical… — JAMB UTME

A rectangular agricultural estate measuring $5\text{ cm}$ by $12\text{ cm}$ on a topographical map represents an actual ground area of $60\text{ km}^2$ . If the map is enlarged such that the estate covers an area of $240\text{ cm}^2$ on the new map, what is the denominator of the Representative Fraction (RF) scale of the enlarged map?

Answer: 50000

Answer

The denominator of the Representative Fraction (RF) scale of the enlarged map is 50000.

The original map area of

60\text{ cm}^2

representing

60\text{ km}^2

gives a linear scale of

1\text{ cm}

1\text{ km}

, which corresponds to an original RF scale of

1 : 100,000

. Enlarging the map area to

240\text{ cm}^2

increases the area by a factor of

4

. The linear enlargement factor is

\sqrt{4} = 2

. Enlarging a map by a linear factor of

2

halves the scale denominator, yielding a new Representative Fraction scale of

1 : 50,000

, with a denominator of

50,000

Step-by-Step Solution

Calculate the area of the estate on the original map.

Original Map Area =

5\text{ cm} \times 12\text{ cm} = 60\text{ cm}^2

Establishes the initial representation of the estate on paper.

Determine the original map scale from the map area and ground area.

Area scale:

60\text{ cm}^2 = 60\text{ km}^2 \implies 1\text{ cm}^2 = 1\text{ km}^2

. Linear scale:

1\text{ cm} = 1\text{ km} = 100,000\text{ cm}

. Original RF =

1 : 100,000

Finding the original scale denominator is necessary before applying the enlargement factor.

Calculate the area enlargement factor.

Area Enlargement Factor =

\frac{240\text{ cm}^2}{60\text{ cm}^2} = 4

Compares the new map area to the original map area.

Calculate the linear enlargement factor (

k

Linear Enlargement Factor

k = \sqrt{4} = 2

Linear scale changes as the square root of the area scale change.

Calculate the new RF scale denominator.

New RF denominator =

\frac{100,000}{2} = 50,000

Enlarging a map linearly by a factor of 2 makes the scale 2 times larger, dividing the denominator by 2.

Key Concept

Map Enlargement and Linear vs Area Scale Conversion

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