After 1 Hour, $\dfrac{1}{8}$ of the initial mass of a certain radioactive isotope remains undecayed. What is the half-life of the isotope?

Hint: To describe the rate at which the isotope will decay and give off radiation, the half-life of an isotope is used. Using the half-life, the amount of radioactive material that will remain after a specified amount of time can be predicted. Calculate Half time by using the relation of mass.

Formula used:
\[HalfTime\propto ~\dfrac{1}{mass}\]

Complete Step-by-Step solution:
Half the original nuclei will disintegrate during the half-life of that substance. Although changing factors such as temperature, concentration, etc have accelerated or slowed chemical changes, these factors do not have an effect on half-life. Each radioactive isotope, independent of any of these factors, will have its own unique half-life.
\[HalfTime\propto ~\dfrac{1}{mass}\]
\[HalfTime\times mass=k\]

Let x be 1st half time and \[0.5\times m\] be half time mass.
Given that after 1-hour mass = \[\dfrac{M}{8}\] or \[0.125M\].
\[\Rightarrow x\times (0.5M)=1\times (0.125M)\]
\[x=\dfrac{0.125\times M}{0.5\times M}\]
\[\therefore x=0.25hr\]
Or
\[\therefore x=15\min \]

The half-life of the isotope is \[15\min \].

Note:
It is important to know about half-lives because it allows you to determine when it is safe to handle a sample of radioactive material. ... They need to be active long enough to treat the condition, but they should also have a half-life that is short enough so that healthy cells and organs are not injured. The half-lives of many radioactive isotopes have been determined and have been found to range from 10 billion years of extremely long half-lives to fractions of a second of extremely short half-lives. For selected elements, the table below illustrates half-lives.