In quantitative cartography, proportional circles are constructed such that the surface ar… — JAMB UTME

In quantitative cartography, proportional circles are constructed such that the surface area of each circle is directly proportional to the statistical quantity being represented. On a demographic map of West Africa, a city with a population of $4,000,000$ is drawn using a proportional circle with a radius of $3.0\text{ cm}$ . Which of the following is the correct radius required to represent a neighboring metropolitan city with a population of $16,000,000$ on the same map?

$6.0\text{ cm}$ Answer
B
$12.0\text{ cm}$
C
$4.0\text{ cm}$
D
$1.5\text{ cm}$

Answer

The correct radius required to represent the city is

6.0\text{ cm}

The area of a proportional circle represents the statistical quantity (

A \propto P

). Because circle area is calculated as

\pi r^2

, the radius

r

must vary with the square root of the population (

r \propto \sqrt{P}

). The population ratio between the two cities is

\frac{16,000,000}{4,000,000} = 4

. Taking the square root gives

\sqrt{4} = 2

. Multiplying the initial radius of

3.0\text{ cm}

2

yields the accurate radius of

6.0\text{ cm}

Step-by-Step Solution

Identify the relationship between statistical data and proportional circle dimensions.

The area of a circle

A = \pi r^2

is proportional to the statistical quantity

P

. Therefore,

r \propto \sqrt{P}

Cartographic principles require symbol areas, not linear radii, to reflect statistical magnitudes.

Set up the ratio between the radii and the square root of populations for both cities.

\frac{r_2}{r_1} = \sqrt{\frac{P_2}{P_1}}

Using ratios eliminates the proportionality constant

k

and simplifies calculation.

Substitute given values into the ratio formula.

\frac{r_2}{3.0} = \sqrt{\frac{16,000,000}{4,000,000}} = \sqrt{4} = 2

Simplifying the population fraction yields a square root factor of 2.

Calculate the unknown radius

r_2

r_2 = 3.0 \times 2 = 6.0\text{ cm}

Multiplying the baseline radius by the scale factor yields the required circle radius.

Key Concept

Square root scaling rule for proportional circle radii in statistical maps

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