A superhero is rendered powerless when exposed to 40 or more grams of a certain

Question

A superhero is rendered powerless when exposed to 40 or more grams of a certain element. A 650 year old rock that originally contained 200 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 300 years. How many grams of the element are still contained in the stolen rock? For how many years can this rock be used by the superhero's nemesis to render the superhero powerless? The stolen rock still contains about grams of the element. (Do not round until the final answer. Then round to two decimal places as needed.)

Explanation / Answer

Here, A0 = 200 gms, A = 100 gms and the half life of the substance being t = 300, we have 100 = 200e-300k or, e-300k = 1/2. On taking natural logarithm of both the sides, we get -300k ln e = ln 1/2 or, -300k = -0.69314718 so that k = 0.69314718/300 = 0.002310490602. Then the present quantity of the substance in the 650 year old rock is 200e(-0.002310490602)*650 = 200e-1.501818891 = 200*0.22272 = 44.544 gms = 44.54 gms ( on rounding off to 2 decimal places).

2. When A = 40 gms, we have 40 = 200 e-0.002310490602t or, e-0.002310490602t = 40/200 = 1/5. On taking natural logarithm of both the sides, we get -0.002310490602t = ln(1/5) = -1.609437912 so that t = 1.609437912/0.002310490602 = 696.58 years (approximately). Since the rock is already 650 years old, it can still be used by the superhero’s nemesis for another 696.58-650 = 46.58 years

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