Decay Rate: The rate $\frac{dA}{dt}at$ which the nuclei $of$ a substance decay $is$ proportional $to$ the

amount $A(t)of$ the substance remaining $at$ time $t$ :

$\frac{dA}{dt}$propA, $or,\frac{dA}{dt}=kA,$ for $,k<mn>0</mn>$

Exercise$-5$: A breeder reactor converts relatively stable uranium$-238$ into the isotope plutonium$-$

After $15$ years $itis$ determined that $0\mathrm{.}043\%of$ the initial amount $A_{0}of$ plutonium has

disintegrated. Find the half$-$life $of$ this isotope $if$ the rate $of$ disintegration $is$ proportional $to$ the

amount remaining.

Exercise$-6$: A fossilized bone $is$ found $to$ contain $0\mathrm{.}1\%of$ its original amount $ofC-14.$ Determine

the age $of$ the fossil, given that the half$-$life $ofC-14is$ approximately $5730$ years.