College Algebra

Question

A superhero is rendered powerless when exposed to 65 or more grams of a certain element. A 550 year old rock that originally contained 300 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 250 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.

Answer 1

(a)

 

t = years elapsed = 550

 

m(0) = initial amount = 300 g

 

m(t) = mass remaining after t years

 

m(t) = m(0) (1/2)^t/250

 

m(t) = (300 g) (1/2)^t/250

 

m(550) = (300 g) (1/2)^550/250 ≅ 65.29 g

 

(b)

 

Set m(t) = 65 g and solve for t, with the initial amount reset to the result from part (a).

 

65 = 65.29 (1/2)^t/250

 

t = 250 ln(65/65.29) / ln(0.5) ≅ 1.606 years