In the extract of Survey of India map G43S7, prepared on a scale of 2 cm to 1 km, a child finds the length of the cart track between two settlements is 7.6 cm. Find :

(a) the actual length of the cart track on the ground.

(b) actual area of a grid square, if each has an area of 4 cm².

Scale factor (k) = 2 cm to 1 km or 2 cm to 100000 cm = $\dfrac{2}{100000} = \dfrac{1}{50000}$

(a) Length of cart track on map = k × Length of cart track on ground

$7.6 = \dfrac{1}{50000}$ × Length of cart track on ground

Length of cart track on ground = 7.6 × 50000 = 380000 cm = $\dfrac{380000}{100000}$ = 3.8 km

Hence, length of actual cart track on ground = 3.8 km

(b) Area of model grid square = k² × Area of actual grid square

⇒ 4 = $\Big(\dfrac{1}{50000}\Big)^2$ × Area of actual grid square

⇒ 4 = $\Big(\dfrac{1}{25 \times 10^8}\Big)$ × Area of actual grid square

⇒ Area of actual grid square = $4 \times 25 \times 10^8$

⇒ Area of actual grid square = 100 × 10⁸ = 10² × 10⁸ = 10¹⁰ cm²

= $\dfrac{10^{10}}{10^{10}}$ km² = 1 km².

Hence, actual area of a grid square = 1 km².