Question

(b) A polluting factory produces emissions whose damage (D) declines with distance. Robin works in this factory. The damage to Robin is D(r)=1000/r, where r is distance in kilometer from the firm and D is damage in rupees. His cost of commuting from the factory to home is ₹20 per km. (i) Write an expression for Robin's total cost. (ii) At what distance from the factory will Robin live? (If he wants to minimize his total cost.) (iii) At what distance from the factory will Robin live if he receives full compensation for the damage he incurs? The minimum distance from the factory where the Robin can live is 0.2 kilometers.

Answer 1

Robin's total cost function is TC(r) = 1000/r + 20r. To minimize his total cost without compensation, Robin should live 5 km away from the factory. If he receives full damage compensation, the optimal distance is the minimum allowed, which is 0.2 km.

Step-by-step explanation:

Robin's Total Cost and Optimal Distance from the Factory

To answer the student's question regarding the polluting factory where Robin works, we can calculate the following:

Robin's total cost (TC) can be expressed as the sum of the damage from pollution (D) and the cost of commuting (C), so TC(r) = D(r) + C(r) = 1000/r + 20r.

1. To find the distance at which Robin will live to minimize his total cost, we take the derivative of TC(r) with respect to r, set it to zero, and solve for r. TC'(r) = -1000/r^2 + 20 = 0. Solving the equation gives r = 5 kilometers. This is the optimal distance to minimize his total cost.
2. If Robin receives full compensation for the damage, the cost function changes since the damage cost D(r) becomes zero effectively, leaving only the commuting cost. The distance in this scenario will be the minimum possible, which is 0.2 kilometers as given by the question constraints.

The student's question involves a practical application of economics in calculating costs associated with both pollution damage and commuting.