Two radioactive materials A and B having decay constant ' 7λ ' and...

Let the initial number of nuclei of both materials be $N_0$.

The number of nuclei of material A at time $t$ is given by $N_A = N_0 e^{-7\lambda t}$.

The number of nuclei of material B at time $t$ is given by $N_B = N_0 e^{-\lambda t}$.

Given that the ratio of the number of nuclei of material B to that of A is $e$:

$\dfrac{N_B}{N_A} = e$

$\dfrac{N_0 e^{-\lambda t}}{N_0 e^{-7\lambda t}} = e$

$e^{6\lambda t} = e^1$

Equating the powers of $e$, we get:

$6\lambda t = 1$

$t = \dfrac{1}{6\lambda}$

Answer: $1/6\lambda$