Probability of a radioactive atom surviving 5 times longer than its half-life

To understand the probability of a radioactive atom surviving 5 times longer than its half-life, we need to consider the concept of half-life and exponential decay.

Half-life:
The half-life of a radioactive substance is the time it takes for half of the initial quantity of the substance to decay. It is a constant characteristic of each radioactive isotope.

Exponential decay:
Radioactive decay follows an exponential decay equation, where the number of radioactive atoms remaining after a certain time is given by N(t) = N0 * (1/2)^(t/T), where N(t) is the number of atoms remaining at time t, N0 is the initial number of atoms, t is the time elapsed, and T is the half-life of the substance.

Now, let's calculate the probability of survival for 5 times longer than the half-life.

Step 1: Calculate the survival probability for one half-life
The probability of an atom surviving one half-life is given by P1 = (1/2)^(1/2) = 0.5^(1/2) = 0.7071

Step 2: Calculate the survival probability for five half-lives
The probability of an atom surviving five half-lives is given by P5 = (1/2)^(5/2) = 0.5^(5/2) = 0.1768

Step 3: Calculate the probability of surviving 5 times longer than the half-life
The probability of an atom surviving 5 times longer than the half-life is given by P = 1 - P5

Substituting the value of P5 into the equation, we get:
P = 1 - 0.1768 = 0.8232

Step 4: Calculate the probability of not surviving 5 times longer than the half-life
The probability of an atom not surviving 5 times longer than the half-life is given by 1 - P = 1 - 0.8232 = 0.1768

Therefore, the probability of a radioactive atom surviving 5 times longer than its half-life is 0.1768.

Conclusion:
The correct answer is not 0.03125 but 0.1768. It seems there might have been a mistake in the given answer. The probability of survival for 5 times longer than the half-life is approximately 0.1768.