Logs/Half-Life/Ratios

Question

A superhero is rendered powerless when exposed to 30 or more grams of a certain element. A 450 year old rock that originally contained 250 grams of this element was recently stolen from a rock museum by the superhero's nemesis. The half-life of the element is known to be 150 years. How many grams of the element are still contained in the stolen rock?

Answer 1

this is a continuous decay problem so we use the formula y= I x e^kt . Where I is initial amt, y is final amount, k is decay constant and t is time. First we need to solve for k. Since the half life is 150 years we have t=150. The ratio of the final to initial is .5 ( that is what half life is)

so .5= e^150k Take the natural log of both sides

ln(.5)= 150 k k= ln(.5)/150= -.00462 we need this k value to solve our problem

Now lets go back to the original scenario. We start with 250 grams and we want to know how much is left in 450 years so

Y= 250 xe^(450 x -.00462)= 31.2638 gm remain

So our superhero will not be so super