A coastal lagoon covers a rectangular section measuring 12\text{ cm} by 15\text{ cm} on Ma… — JAMB UTME

A coastal lagoon covers a rectangular section measuring $12\text{ cm}$ by $15\text{ cm}$ on Map A, which is drawn to a scale of $1 : 20,000$ . If Map A is reduced to create Map B with a scale of $1 : 60,000$ , what is the area of the lagoon on Map B in square centimeters?

Answer: 20 cm²

Answer

The area of the lagoon on Map B is

20\text{ cm}^2

Reducing the scale from

1 : 20,000

1 : 60,000

reduces all linear dimensions to

\frac{1}{3}

of their original length. Consequently, the area changes by

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

. Taking the original map area of

180\text{ cm}^2

(

12\text{ cm} \times 15\text{ cm}

) and multiplying by

\frac{1}{9}

gives

20\text{ cm}^2

Step-by-Step Solution

Calculate the surface area of the lagoon on the original map (Map A)

Area on Map A =

12\text{ cm} \times 15\text{ cm} = 180\text{ cm}^2

Determining the initial area on paper establishes the base value before scale reduction.

Determine the linear scale reduction ratio

Linear scale factor =

\frac{\text{Original Scale Denominator}}{\text{New Scale Denominator}} = \frac{20,000}{60,000} = \frac{1}{3}

Increasing the scale denominator from 20,000 to 60,000 means linear distances shrink to one-third of their original length.

Compute the area scale conversion factor

Area scale factor =

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

Map area varies as the square of the linear scale ratio.

Calculate the final reduced area on Map B

New area on Map B =

180\text{ cm}^2 \times \frac{1}{9} = 20\text{ cm}^2

Multiplying the original map area by the area scale factor gives the resulting map area.

Key Concept

When a map scale is reduced, linear dimensions change by the factor

k = \frac{\text{Old Denominator}}{\text{New Denominator}}

, while the map area changes by the factor

k^2

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