Word problesm including Half-life

Question

A superhero is rendered powerless when exposed to 55 or more grams of a certain element. A 350 year old rock that originally contained 650 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 100 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?

 

I think part of my problem is that all the extra "superhero" information is throwing me off a bit. But if someone could just help me get started and explain to me what formulas to use, that would be great.

Answer 1

A = a₀(0.5)^t/h

A = 650(0.5)^350/100

A = 650(0.5)^3.5

A ≈ 650(0.088388)

A ≈ 57.4524

 

The hero should be on guard!

 

55 = 650(0.5)^t/100

0.0846 ≈ (0.6)^t/100

ln 0.0846 ≈ (t/100) ln 0.5

-2.4698 ≈ (t/100) (-0.6931)

-246.98 ≈ t (-0.6931)

356.34 ≈ t

 

Hero should hide out for another 6.34 years!

Answer 2

This may be solved as an exponential decay problem, where the mass of the remaining element, as a function of time, is given by

 

m(t) = m(0)e^-kt

 

where m(t) is in grams, m(0) is the initial mass, t is years, and k is a constant to be determined from the given half-life of 100 years.

 

t_1/2 = 100 years

 

Therefore,

 

0.5  = e^-k(100) 

 

k = ln(0.5) / (-100) years^-1 = 0.006931 years^-1

 

Given m(0) = 650 grams,

 

m(t) = 650 e^-0.006931t grams

 

(a)

 

At 350 years,

 

m(350) = 650 e^{-0.006931(350)} = 57.46 grams

 

(b)

 

In this part of the problem, we need to know how long it takes for the 57.46 grams to decay to 55 grams.  Then 57.46 grams becomes the initial mass in this part of the problem.

 

55 = 57.46 e^-0.006931t

 

t = ln(55 / 57.46) / (-0.006931) years = 6.313 years

 

The 350 year old rock will remain dangerous for another 6.313 years.