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**Hint:**The radioactive decay constant is defined as the probability of a given unstable nucleus decaying per unit time. It is denoted by $\lambda$.

**Complete step by step answer:**

(a) We know that the activity of the sample is given as,

$ R\quad =\quad { R }_{ 0 }{ e }^{ -\lambda t }$

where $\lambda$ is the decay constant, ${R}_{0}$ is the activity at time 0 and R is the activity at time t.

$\implies \cfrac { R }{ { R }_{ 0 } } \quad =\quad { e }^{ -\lambda t }$

Now, taking natural log both sides we get,

$ ln(\cfrac { R }{ { R }_{ 0 } } )\quad =\quad -\lambda t$

$\implies ln(\cfrac { { R }_{ 0 } }{ { R } } )\quad =\quad \lambda t$

$\implies \lambda \quad =\quad \cfrac { 1 }{ t } ln(\cfrac { { R }_{ 0 } }{ { R } } )$

Now converting the natural log to base 10 we get,

$ \lambda \quad =\quad \cfrac { 2.303 }{ t } log(\cfrac { { R }_{ 0 } }{ { R } } )$ -----(1)

Now, it is given in the question that t=5 mins, Activity at time 0, ${R}_{0}$ is 4750 count/minute and Activity at time t, R is 2700 count/minute. Substituting the value in equation (1) we get,

$ \lambda \quad =\quad \cfrac { 2.303 }{ 5 } log(\cfrac { 4750 }{ { 2700 } } )$

$\implies \lambda \quad =\quad \cfrac { 2.303 }{ 5 } log(1.76)$

$\implies \lambda \quad =\quad \cfrac { 2.303 }{ 5 } \quad \times \quad 0.2455$

$\implies \lambda \quad =\quad 0.1131/min$

Therefore, the decay constant is 0.1131/minute.

(b) Now, we know that the mean life of the sample is denoted by $\tau$ and is given as the reciprocal of the decay constant.

$ \tau \quad =\quad \cfrac { 1 }{ \lambda }$

And we found the value of $\lambda$ as 0.1131/min. Substituting the value of decay constant in the above equation, we get

$ \tau \quad =\quad \cfrac { 1 }{ 0.1131 }$

$\implies \tau \quad =\quad 8.85\quad min$

Now, half-life of the sample is given as,

$ T\quad =\quad 0.693\quad \times \quad \tau$

Substituting the value of $\tau$, we get

$ T\quad =\quad 0.693\quad \times \quad 8.85$

$\implies T\quad =\quad 6.13\quad min$

Therefore, the half-life of the sample is 6.13 minutes.

**Note:**While calculating the decay constant, do make sure that the base of the log is 10 else you might end up getting the wrong answer. The half-life of a sample is 69.3% of the mean life. It is applicable for any sample.

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