Question

A superhero is rendered powerless when exposed to 3535 or more grams of a certain element. A 550550 year old rock that originally contained 250250 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 200200 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

Answer:

(a)

The quantity of element that are still contained in the stolen rock

is 37.1627g

(b)

this rock can be used by the superhero's nemesis to render the superhero powerless is 567 years

Step-by-step explanation:

We are given

original quantity =250g

half life time =200 years

so, we can use formula

$P(t)=P_0(0.5)^{\frac{t}{h} }$

where

Po is initial quantity

h is half life time

t is time in years

so, we can plug h=200  and Po=250

and we get

$P(t)=250(0.5)^{\frac{t}{200} }$

(a)

we can plug t=550

and then we can find P(t)

$P(550)=250(0.5)^{\frac{550}{200} }$

we get

$P(550)=37.1627g$

(b)

we can set P(t)=35

and then we can solve for t

$35=250(0.5)^{\frac{t}{200} }$

$0.5^{\frac{t}{200}}=\frac{7}{50}$

we can take ln on both sides

$\ln \left(0.5^{\frac{t}{200}}\right)=\ln \left(\frac{7}{50}\right)$

$t=\frac{200\ln \left(\frac{7}{50}\right)}{\ln \left(0.5\right)}$

$t=567.3$

$t=567$ years