The base of a triangular field is three times its altitude. If the cost of cultivating the field is ₹24.68 per hectare and the total cost paid is ₹333.18, find the base and altitude of the field (in metres). Assume 1 hectare = 10,000 m^2.

Introduction / Context: This problem combines unit conversion (hectares to square metres), cost-to-area calculation, and triangle area relationships. The key idea is: total cost and rate per hectare tell us the area of the field, and the geometry relationship (base = 3*altitude) lets us solve for the dimensions.

Given Data / Assumptions:

• Cost rate = ₹24.68 per hectare
• Total cost = ₹333.18
• 1 hectare = 10,000 m^2
• Base b = 3h where h is altitude
• Area of triangle A = (1/2) * b * h

Concept / Approach: First compute area in hectares using Area(hectares) = Total cost / Rate. Convert it to m^2. Then use b = 3h in the triangle area formula to create an equation in h, solve for h, and get b.

Step-by-Step Solution: Area in hectares = 333.18 / 24.68 = 13.5 hectares Convert to m^2: A = 13.5 * 10,000 = 135,000 m^2 Given b = 3h Triangle area: A = (1/2) * b * h = (1/2) * (3h) * h = (3/2) * h^2 = 1.5*h^2 So 1.5*h^2 = 135,000 h^2 = 135,000 / 1.5 = 90,000 h = sqrt(90,000) = 300 m b = 3h = 3*300 = 900 m

Verification / Alternative check: Using b = 900 and h = 300, area = (1/2)*900*300 = 135,000 m^2 = 13.5 hectares. Cost = 13.5*24.68 = ₹333.18, exactly correct.

Why Other Options Are Wrong: Base = 300; Altitude = 900: violates base = 3*altitude. Base = 800; Altitude = 350: base is not 3 times altitude and area would not match 135,000 m^2. Base = 600; Altitude = 400: base is not 3 times altitude and area differs. Base = 750; Altitude = 250: base = 3*250 is 750 (ratio fits), but area = (1/2)*750*250 = 93,750 m^2, not 135,000 m^2.

Common Pitfalls: Not converting hectares to m^2. Using rectangle area instead of triangle area. Mixing up the relation and taking altitude as 3 times base.

Final Answer: Base = 900 m and Altitude = 300 m