The base of a triangular field is three times its height. If the cost of cultivating the field at ₹ 36.72 per 100 m² is ₹ 49,572; find its base and height.

Given:

Cost of cultivating the field = ₹ 36.72 per 100 m²

Total cost = ₹ 49,572

Total cost = Area x Cost of cultivating per 100 m²

Area = $\dfrac{\text{Total cost}}{\text{Cost of cultivating}}$

$= \dfrac{49,572 \times 100}{36.72}\\[1em] = \dfrac{49,572 \times 10,000}{3672}\\[1em] = 1,35,000 \text{ m}^2$

Let the height of the triangle be h.

Since the base of the field is three times its height, we have:

Base = 3h

Area = $\dfrac{1}{2}$ x base x height

$⇒ 1,35,000 = \dfrac{1}{2} \times 3h \times h\\[1em] ⇒ 1,35,000 = \dfrac{3}{2} \times h^2\\[1em] ⇒ h^2 = \dfrac{1,35,000 \times 2}{3}\\[1em] ⇒ h^2 = \dfrac{2,70,000}{3}\\[1em] ⇒ h^2 = 90,000\\[1em] ⇒ h = \sqrt{90,000}\\[1em] ⇒ h = 300$

Thus, the height of the triangle is 300 m.

Base = 3 x height = 3 x 300 m = 900 m

Hence, the height is 300 m and the base is 900m.