1) Praful will minimize the cost of travelling to and fro from the airport as well as the damage caused due to noise:
Cost of travelling= Rs. 2d , (Since the cost is 1 rupee per km and he has to travel distance d twice- to and fro)
Damage caused due to noise in terms of rupees= 1* N(d)= 1/d^2
Thus total cost to him for living at distance d from airport= 2d + 1/d^2
minimizing total cost by differentiating with respect to d: dT.C/dd = 2-2/d^3
and equating to zero : 2=2/d^3 => d^3=1=> d=1.
2) Since praful is compensated for the damage caused by the noise he would prefer to only minimize the cost of travelling this means he will live as close as possible to the airport. Since it is given that the closest he can live to the airport is d=.1 km therefore the compensation he would receive= 1* N(0.1)= 1/(0.1)^2 = Rs. 100.
@Ram @Ayushya It is given that: J+R=96
Just maximize Juliet's utility function :
U=R^4J^8 = (96-J)^4 J^8
dU/dJ= 4 (-1)(96-J)^3 J^8 + (96-J)^4 (8) J^7=0
When you guys solve this equation J=64 and R=32. This clearly => d) Juliet would want to give Romeo some chocolates if she had more than 64.