A refinery needs to supply 50 barrels of petrol at the end of the first month, 80 barrels at the end of second month, and 100 barrels at the end of third month. The production cost of $x$ barrels of petrol in any month is given by $\$\left(450 x+20 x^{2}\right)$. It can produce more petrol in any month and supply it in the next month. However, there is an inventory carrying cost of $\$ 10$ per barrel per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.