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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 150 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 b 200 years .

a) How many grams of the element are still contained in the stolen rock?

b) For how many years can this rock be used by the superhero's nemesis to render the superhero powerless ?

A) The stolen rock still contains about_____grams of the element.

round to two decimal places as needed

Answer 1

If only 1/2 of the material remains after 200 years then we could create an equation that multiplied by 1/2 every 200 years: Multiply the initial amount by (1/2)^t/200 where "t" is the amount of years that go by.

Like this: A(t) = A₀(1/2)^t/200

After 200 years we multiply the initial amount (A₀) by (1/2)^200/200 or by 1/2 and after 400 years, we multiply the initial amount by (1/2)^400/200 or (1/2)²

So if A₀ = 150, the equation becomes A(t) = 150(1/2)^t/200 and if t = 450 then:

A(450) = 150(1/2)^450/200

A(450) = 150(1/2)^2.25

A(450) = 150(0.210224) = 31.53 grams

For part b, we need to find "t" is the amount is 30 grams and this will be our boundary. Everything smaller than that is OK (so we want the time to be longer than the time it takes to get to 30). To find the time it takes to get to 30, we set A(t) equal to 30:

30 = 150(1/2)^t/200

30/150 = (1/2)^t/200

0.2 = (1/2)^t/200

log(0.2) = log[(1/2)^t/200]

-0.69897 = (t/200)log(0.5)

-0.69897 = (t/200)(-0.30103)

-0.69897/-0.30103 = (t/200)

2.32193 = t/200

t = (200)(2.32193) = 464.39 years so any time after 464.39 years there will be less than 30 grams of the element left.