The altitude and the base of a triangular field are in the ratio 6 : 5. If its cost is ₹ 49,57,200 at the rate of ₹ 36,720 per hectare and 1 hectare = 10,000 sq.m, find (in metre) the dimensions of the field.

Given:

Ratio of altitude and base of the field = 6 : 5

Total cost = ₹ 49,57,200

Rate = ₹ 36,720 per hectare

As we know that area of the triangular field x rate = total cost

⇒ Area of the triangular field x $\dfrac{₹ 36,720}{1,00,00}$ = ₹ 49,57,200

⇒ Area of the triangular field = ₹ $\dfrac{49,57,200}{36,720} \times 10,000$

⇒ Area of the triangular field = ₹ $135 \times 10,000$

⇒ Area of the triangular field = 13,50,000 m²

Let the altitude be 6a and the base be 5a.

As we know, the area of a triangle = $\dfrac{1}{2}$ x base x height

⇒ 13,50,000 = $\dfrac{1}{2}$ x 6a x 5a

⇒ 13,50,000 = $\dfrac{1}{2}$ x 30a²

⇒ 13,50,000 = 15a²

⇒ a² = $\dfrac{13,50,000}{15}$

⇒ a² = 90000

⇒ a = $\sqrt{90000}$

⇒ a = 300

So, the altitude = 6a

= 6 x 300

= 1800 m

And, the base = 5a

= 5 x 300

= 1500 m

Hence, the dimensions of the field are 1800 m and 1500 m.