Question

$A$ and $B$ are two substances undergoing radioactive decay in a container The half life of $A$ is $15 min$ and that of $B$ is $5\, min$ If the initial concentration of $B$ is $4$ times that of $A$ and they both start decaying at the same time, how much time will it take for the concentration of both of them to be same? ______ min

Answer 1

The correct answer is 15.
[A]t​=[A]0​e−kt
For A : Let [A]t​ be y and [A]0​ be x;k=t1/2​ln2​=
15minln2​
y=xe−kt
=xe−(15ln2​)t
For B: [B]t​=[B]0​e−kt
Let [B]t​=y;[B]0​=4x;k=t1/2​ln2​=5minln2​
y=4xe−(5ln2​)t
⇒xe−(15ln2​)t=4xe−(5ln2​)t
et(5ln2​15ln2​)=4
t×[5ln2​−15ln2​]=ln4
t×ln2[51​−151​]=2ln2
t=15min