Question

(Radioactivity) The half-life of radioactive cobalt is 5.3 years. After a nuclear reactor accident, the surrounding region had 100 times the amount of cobalt acceptable for habitation. How long will it be before the region is habitable?

Answer 1

Answer: It will be 35 years before the region is habitable.

Step-by-step explanation:

Since we have given that

Half life of radioactive cobalt = 5.3 years

Let the initial amount of cobalt = x

final amount of cobalt = 100x

So, we need to find the time that before the region is habitable.

So, it becomes,

$N=N_0(\dfrac{1}{2})^{\frac{t}{t_{\frac{1}{2}}}}\\\dfrac{N}{N_0}=(0.5)^{\frac{t}{t_{\frac{1}{2}}}}\\\dfrac{100x}{x}=0.5^{\frac{t}{5.3}}\\\\100=0.5^{\frac{t}{5.3}}\\\\\text{Taking log on both sides}\\\\\log 100=\dfrac{t}{5.3}\log 0.5\\\\\dfrac{\log 100}{\log 0.5}=\dfrac{t}{5.3}\\\\6.643\times 5.3=t\\\\35.212=t\\\\t\approx 35\ years$

Hence, it will be 35 years before the region is habitable.