Answer to Question #121809 in Calculus for Servan

Question

Silver ore was found in a region. Then, as a result of the investigations, silver reserves in the region were estimated to be 50000 tons. Later, a mine was established in the region and silver reserves in the region began to be mined. According to the amount of silver extracted from the mine, e (-t / 40000), the rate of change depending on time t time. Under the assumption that the technology will not change, find the number of tons of silver that cannot be removed from this mine.

Answer 1

Let x be the amount of silver ore reserved in the mine at time t.

So $\frac {dx}{dt}$ = - $e^{-\frac {t}{40000}}$ [ -ve sign is taken as

ore decreasing]

=> dx = - $e^{-\frac {t}{40000}}$ dt

Integrating ,

$\int dx = - \int e^{-\frac {t}{40000}}dt$

=> x = 40000 $e^{-\frac {t}{40000}}$ + C , C is

integration constant

By initial condition , t = 0, x = 50000

50000 = 40000 + C

=> C = 10000

So amount of silver reserved in mine after time t is given by

x = 40000$e^{-\frac {t}{40000}}$ + 10000

As t →∞ , $e^{-\frac {t}{40000}}$ →0

and 40000$e^{-\frac {t}{40000}}$ →0

so x →10000 as t→∞

So 10000 tons silver can not be removed from mine