Problem $4(15$ points$)$

A gold mine is for lease for three years. It is estimated that the current

amount of gold remaining in the mine is $x_0=30,000$ ounces. The price of gold

is currently $$1000$ per ounce. If $x$ is the amount of gold remaining in the mine

at the start of a year, the cost to extract $$500\frac{z^2}{x}5\%z$ ounces $of$ gold $in$ that year $is$

$$500\frac{z^2}{x}.$ The interest rate $is5\%.$ How much $is$ this lease worth? What $is$ the

optimal extraction amount $at$ the start $of$ each year? Show all work including

your work taking derivatives.