A river basin covers a rectangular section measuring 4.5\text{ cm} by 8\text{ cm} on Map X… — JAMB UTME

A river basin covers a rectangular section measuring $4.5\text{ cm}$ by $8\text{ cm}$ on Map X, which has a representative fraction of $1 : 25,000$ . Map X is enlarged to create Map Y, where the same river basin occupies an area of $144\text{ cm}^2$ . Map Y is then reduced to construct Map Z, such that the denominator of Map Z's representative fraction is $4$ times that of Map Y. What is the area of the river basin on Map Z in $\text{cm}^2$ ?

Answer: 9 cm^2

Answer

The area of the river basin on Map Z is

9\text{ cm}^2

To find the area on Map Z, first calculate the basin's area on Map X:

4.5\text{ cm} \times 8\text{ cm} = 36\text{ cm}^2

. The area ratio from Map X to Map Y is

144 / 36 = 4

, which corresponds to a linear enlargement factor of

\sqrt{4} = 2

. Map Y therefore has a scale of

1 : 12,500

. Multiplying Map Y's RF denominator by

4

gives Map Z a scale of

1 : 50,000

. Comparing Map Y (

1 : 12,500

) to Map Z (

1 : 50,000

) reveals a linear reduction factor of

1/4

. The area reduction factor is

(1/4)^2 = 1/16

. Multiplying Map Y's area by

1/16

yields

144\text{ cm}^2 \times (1/16) = 9\text{ cm}^2

. Alternatively, comparing Map X (

1 : 25,000

) directly to Map Z (

1 : 50,000

) shows a linear factor of

1/2

, giving an area factor of

(1/2)^2 = 1/4

, and

36\text{ cm}^2 \times (1/4) = 9\text{ cm}^2

Step-by-Step Solution

Calculate the initial map area on Map X.

A_X = 4.5\text{ cm} \times 8\text{ cm} = 36\text{ cm}^2

The product of the rectangular map dimensions gives the original map area.

Determine the linear scale factor and representative fraction of Map Y.

Area enlargement factor =

144 / 36 = 4

; Linear enlargement factor =

\sqrt{4} = 2

. Map Y scale =

1 : (25,000 / 2) = 1 : 12,500

Area change ratio is the square of the linear scale factor (

n^2

). Enlarging a map decreases its RF denominator by the linear factor.

Calculate the representative fraction of Map Z.

Map Z RF denominator =

12,500 \times 4 = 50,000

, so scale of Map Z =

1 : 50,000

Reducing a map increases the RF denominator proportionately.

Determine the final map area on Map Z.

Linear reduction factor from Y to Z =

12,500 / 50,000 = 1/4

. Area reduction factor =

(1/4)^2 = 1/16

. Final area

A_Z = 144\text{ cm}^2 \times (1/16) = 9\text{ cm}^2

Applying the squared linear scale reduction factor to Map Y's area yields Map Z's area.

Key Concept

Map Enlargement, Reduction, and Area-to-Linear Scale Ratio Relationships

Estimated Time:3m 0s

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