哈密尔顿函数的问题

Fora mining firm, the cost function is C=0.01q(t)2, theprice of product p=10，thediscount rate is 0.2. The initial stock of mine is 1, i.e, X（0）=1.Assuming the mine will be completely extracted with one period, i.e., X（1）=0，whatis the optimal time path, X*（t）, for the firm?
各位高人能不能帮我解解这道题，拜托了

What do you mean by optimal time path?
I think the problem can be solved by Monte Carlo simulation as well.

就是推导出一个x关于t的函数，
那如果用Monte Carlo simulation，能否再详细说一下具体过程