Half life word problem

Question

A superhero is rendered powerless when exposed to 40 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 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?

Answer 1

300 grams 550 years ago when t=0

half life = 200 years

150 grams 350 years ago when t=200

75 grams 150 years ago when t=400

37.5 grams 50 years from now when t=600

find grams now when t=550

and when will grams decay to 40 grams

A=Pe^rt

A/P =150/300 = 1/2 = e^200r

200r = ln.5

r = (ln.5)/200

A=300e^(ln.5)/200)(550) = Amount now

= 300e^(11/4)ln.5)= 300e^(2.75)(ln.5)

= about 44.60 grams now

40 = 300e^(t/200)(ln.5), solve for t, to find when grams =40

ln(2/15) = (t/200)ln.5

t = 200ln(2/15)/ln.5

= about 581.38 = 31.38 years from now

is last time the nemesis can render superhero powerless