Cost of canal lining: If a canal has total length L (in km), perimeter of lining P (in m), and lining cost C (₹ per m²), what is the additional expenditure on providing lining?

Introduction / Context:

Lining a canal reduces seepage and enhances hydraulic efficiency, but it incurs a capital cost proportional to the area lined. For a prismatic canal, the lined area equals the plan length of the canal multiplied by the wetted perimeter being lined. Converting units consistently is the key to a correct cost estimate.

Given Data / Assumptions:

• Total canal length L is given in kilometres.
• Perimeter of lining P is given in metres (the wetted perimeter or specified portion to be lined).
• Lining cost C is given per square metre.
• Continuous lining along the full length is assumed.

Concept / Approach:

Area to be lined A = (length in metres) * (perimeter in metres). Since L is in km, convert to metres by multiplying with 1000. Thus A = (1000 * L) * P. Total lining cost = A * C.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: L (km) → m, so 1000 * L is in metres. Multiplying by P (m) gives area (m²). Multiplying by C (₹/m²) gives ₹, as required.

Why Other Options Are Wrong:

• L * P * C: Misses the km→m conversion factor.
• (P / L) * C or (L * C) / P: Do not represent an area; wrong dimensions.
• P * C / 1000: Inverts the conversion; underestimates the cost drastically.

Common Pitfalls:

• Forgetting the unit conversion from km to m.
• Using wetted perimeter at one depth while costing a different lining extent; ensure P matches the lined profile.

Final Answer:

1000 * L * P * C