The Table X above shows the number of farm units in three villages of Oju District in the year 2010. Use the data in Table X to answer the questions that fo...

Data read from Table X

(a) Constructing the proportional circles

In a proportional circle the area of the circle (not the radius) is made proportional to the quantity it represents. Since the area of a circle is \( A = \pi r^{2} \), the area is proportional to \( r^{2} \). Therefore, to make the area proportional to the number of farm units \( N \), the radius must be proportional to the square root of \( N \):

\[ A \propto N \quad\Rightarrow\quad \pi r^{2} \propto N \quad\Rightarrow\quad r \propto \sqrt{N} \]

Applying the given scale. The scale states that a radius of \(1\ \text{cm}\) represents \(3\) farm units. Taking the reference circle of radius \(1\ \text{cm}\) to stand for \(3\) farm units by its area, the area per farm unit is fixed, so:

\[ \pi r^{2} = \frac{N}{3}\,\pi \quad\Rightarrow\quad r = \sqrt{\frac{N}{3}} \ \text{cm} \]

Working for each village:

• Village A: \( r_{A} = \sqrt{\dfrac{81}{3}} = \sqrt{27} = 5.2\ \text{cm} \)
• Village B: \( r_{B} = \sqrt{\dfrac{100}{3}} = \sqrt{33.3} = 5.8\ \text{cm} \)
• Village C: \( r_{C} = \sqrt{\dfrac{64}{3}} = \sqrt{21.3} = 4.6\ \text{cm} \)

Notice that because the given values are perfect squares, the radii keep the neat ratio \( \sqrt{81}:\sqrt{100}:\sqrt{64} = 9:10:8 \), so village B has the largest circle, village A the middle one and village C the smallest.

Table of radii for drawing:

Steps to construct on the map of Oju District:

1. Locate the position of each village (A, B and C) on the outline map provided.
2. Using a pair of compasses, set the radius to the calculated value and draw a circle centred exactly on each village’s position: A = \(5.2\ \text{cm}\), B = \(5.8\ \text{cm}\), C = \(4.6\ \text{cm}\).
3. Shade or lightly tint each circle uniformly and label it with the village letter and its value.
4. Provide a key/legend stating the scale (\(1\ \text{cm}\) radius represents \(3\) farm units) and give the map a clear title.

(b) Two disadvantages of proportional circles

• They are difficult to draw accurately: the radius of each circle must first be found from the square root of the value, so the calculation and the precise drawing of the correct radius are time-consuming and error-prone.
• Actual values cannot be read directly: the human eye cannot judge the area of a circle accurately, so a reader can only estimate, and one is forced to rely on written labels rather than reading the quantity from the size of the circle. (Large circles may also overlap and crowd nearby locations, hiding detail on the map.)