Question

If the half-life of a radioactive element is 4 days, how long will it take for three- fourths of a sample of the element to decay?

Answer 1

`\boxed{\text{8 da}}`

Step-by-step explanation:

The question will be easier to solve if we interpret it as, " How long will it take until one-fourth of a sample of the element remains,?"

The half-life of the element is the time it takes for half of it to decay.

After one half-life, half (50 %) of the original amount will remain.

After a second half-life, half of that amount (25 %) will remain, and so on.

We can construct a table as follows:

`\begin{array}{cccl}\textbf{No. of} & & \textbf{Fraction} & \\\textbf{half-lives} & \textbf{t/da} & \textbf{remaining} & \\1 & 4 & (1)/(2) & \\\\2 & 8 & (1)/(4)& \\\\3 & 12 & (1)/(8)& \\\end{array}`

`\text{We see that 8 da is two half-lives, and the fraction of the element remaining is $(1)/(4)$.}\\\text{It takes $\boxed{\textbf{8 da}}$ for three-fourths of the element to decay}`