A topographical map with a scale of 1 : 100,000 depicts a reservoir that covers an area of… — JAMB UTME

A topographical map with a scale of $1 : 100,000$ depicts a reservoir that covers an area of $16\text{ cm}^2$ on the map. If the map is reduced to a new scale of $1 : 250,000$ , what is the area of the reservoir on the new reduced map in square centimeters ( $\text{cm}^2$ )?

Answer: 2.56 cm²

Answer

The area of the reservoir on the new map is

2.56\text{ cm}^2

To calculate the new map area after reduction, first find the linear scale factor

k = \frac{100,000}{250,000} = 0.4

. Because area changes in proportion to the square of linear dimensions, the area scale factor is

(0.4)^2 = 0.16

. Multiplying the original map area (

16\text{ cm}^2

) by

0.16

gives the correct new area of

2.56\text{ cm}^2

Step-by-Step Solution

Determine the linear scale factor (

k

)

k = \frac{\text{Original Scale Denominator}}{\text{New Scale Denominator}} = \frac{100,000}{250,000} = 0.4

Map reduction changes linear dimensions proportionally to the ratio of original scale denominator to new scale denominator.

Calculate the area scale factor (

k^2

)

k^2 = (0.4)^2 = 0.16

Map area varies with the square of the linear scale factor.

Compute the new map area

\text{New Area} = \text{Original Area} \times k^2 = 16\text{ cm}^2 \times 0.16 = 2.56\text{ cm}^2

Applying the area scale factor to the original map area gives the reduced surface area on the new map.

Key Concept

Map Reduction and Area Scale Transformation

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