Problem $1$:

A mining company is evaluating a section of a gold deposit to decide whether it is economically viable to develop a new open$-$pit mine. The section under consideration is a rectangular block measuring $500$ meters in length, $200$ meters in width, and $150$ meters in depth. The gold distribution within this block is known to be log$-$normally distributed due to extensive sampling and analysis. Estimate the recoverable reserves of gold in this block using a cut$-$off grade strategy and determine if the section is economically viable for mining operations. Assume the mean gold grade of the block is $2$ g$/$t $($grams per ton$)$ with a standard deviation of $0\mathrm{.}5$ g$/$t$.$

$$ Calculate the cut$-$off grade for gold mining in this deposit, considering the gold price is $$1,500$ per ounce, mining cost is $$25$ per ton of ore, processing cost is $$15$ per ton of ore, and the mine's operational cost is $$10$ million per year $(1$ troy ounce $=31\mathrm{.}1035$ grams$).$

$$ Determine the tonnage of ore above the cut$-$off grade by integrating the grade$-$tonnage curve derived from the block's gold grade distribution.

$$ Estimate the total amount of recoverable gold in ounces using the cut$-$off grade and the tonnage of ore above this grade.

$$ Calculate the gross revenue from the recoverable gold.

$$ Deduct the total mining, processing, and operational costs from the gross revenue to determine the net profit or loss.

Assume a mining efficiency of $90\%$ and a metallurgical recovery rate of $95\%.$

$($Note: Make suitable assumptions for any missing data. Cite references for your assumptions$)$

Make sure every calculation is written down.