Question

6 Suppose a Gaussian curve is used to approximate the production of chromium. If production peaks t six times its current rate of 2 million tons per year, how long would it take to reach that maximum if the total chromium ever mined is 4 billion tons (five times the current reserves)? How long would it take to consume about 80 percent of the total resource?

Answer 1

Given that Ultimate total chromium = 4 billion tons Q_infinity = 4 Billion tons/years Q_infinity = 4000 M tons/yr and Production packs at six times its current rate of 2 million tons per year i.e., P_0 = 2 M tons/year (Peak) P_M = 12 M tons/year We know that Total amount available [for above the mean] sigma = Q_infinity/p_n squareroot 2 pi = 4000 1000/12 3 squareroot 2 pi sigma EqualTilde 133 years Therefore from Probability table [Gaussian] we know that 80% of chromium will be used at a point of 0.84 sigma above the mean, so 0.84 times 133 = 112 years past the mean \r\n we have to find time from now reach the point so t_m = sigma squareroot 2 l_n p_m/p_0 t_m = sigma squareroot 2 ln 12 M Ton/year/2 M Ton/year = 133 squareroot 2ln 12M/2M t_m = 252 years Therefore The time from now to reach the point is therefore