A topographical map, Map A, drawn to a scale of 1 : 100,000, displays a rectangular reserv… — JAMB UTME

A topographical map, Map A, drawn to a scale of $1 : 100,000$ , displays a rectangular reservoir measuring $5\text{ cm}$ by $8\text{ cm}$ . When Map A is reduced to create Map B, the reservoir covers an area of $10\text{ cm}^2$ on Map B. What is the representative fraction (R.F.) scale of Map B?

A
$1 : 50,000$
$1 : 200,000$ Cevap
C
$1 : 400,000$
D
$1 : 25,000$

Cevap

The representative fraction scale of Map B is

1 : 200,000

The area of the reservoir on Map A is

5\text{ cm} \times 8\text{ cm} = 40\text{ cm}^2

. On Map B, the area is reduced to

10\text{ cm}^2

, giving an area reduction ratio of

\frac{10}{40} = \frac{1}{4}

. Since area scale change is the square of the linear scale change (

n^2

), the linear reduction factor is

\sqrt{\frac{1}{4}} = \frac{1}{2}

. Reducing a map by a linear factor of 2 doubles its scale denominator from

100,000

200,000

, resulting in a new scale of

1 : 200,000

Adım Adım Çözüm

Calculate the area of the reservoir on Map A.

\text{Area on Map A} = 5\text{ cm} \times 8\text{ cm} = 40\text{ cm}^2.

The surface area of a rectangular section on a map is obtained by multiplying length by width.

Determine the area reduction factor between Map A and Map B.

\text{Area Ratio} = \frac{\text{Area on Map B}}{\text{Area on Map A}} = \frac{10\text{ cm}^2}{40\text{ cm}^2} = \frac{1}{4}.

Dividing the area on the new map by the area on the original map gives the factor of areal change.

Calculate the linear reduction factor.

\text{Linear Ratio} = \sqrt{\text{Area Ratio}} = \sqrt{\frac{1}{4}} = \frac{1}{2}.

The change in surface area follows the square of the linear scale change (

n^2

), so taking the square root gives the linear factor (

n

Calculate the new Representative Fraction (R.F.) for Map B.

\text{New Scale Denominator} = 100,000 \times 2 = 200,000

, yielding a scale of

1 : 200,000$.

Map reduction results in a smaller scale, which increases the scale denominator proportionally to the linear reduction factor.

Anahtar Kavram

Map Reduction and Area Scale Relationship

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