Step 1: Understanding the Question:
This question deals with nuclear physics, specifically focusing on the law of radioactive decay and half-life.
We are given the half-life of a radioactive sample (10 days) and need to calculate the fraction of the initial nuclei that have decayed after an elapsed time of 40 days.
Step 2: Key Formulas and Approach:
Radioactive decay follows first-order kinetics. The fraction of initial nuclei remaining undecayed ($N/N_0$) after $n$ half-lives is:
\[
\frac{N}{N_0} = \left(\frac{1}{2}\right)^n
\]
Where $n$ is the number of half-lives, calculated as:
\[
n = \frac{\text{Total time } (t)}{\text{Half-life } (T_{1/2})}
\]
The fraction of the initial nuclei decayed is then given by:
\[
\text{Fraction decayed} = 1 - \frac{N}{N_0}
\]
Step 3: Detailed Explanation:
Identify the given parameters: The half-life is $T_{1/2} = 10\text{ days}$, and the total time elapsed is $t = 40\text{ days}$.
Calculate the number of half-lives ($n$):
\[
n = \frac{40}{10} = 4
\]
This means the sample has undergone exactly four half-life cycles.
Determine the fraction of remaining nuclei: After 4 half-lives, the fraction of active nuclei that remain undecayed is:
\[
\frac{N}{N_0} = \left(\frac{1}{2}\right)^4 = \frac{1}{16}
\]
Calculate the fraction of decayed nuclei: The fraction of nuclei that have decomposed is the difference between the initial amount (1) and the remaining undecayed fraction:
\[
\text{Fraction decayed} = 1 - \frac{N}{N_0} = 1 - \frac{1}{16} = \frac{15}{16}
\]
This calculation shows that $15/16$ of the original sample has decayed, leaving only $1/16$ of the original active nuclei intact.
Step 4: Final Answer:
The fraction of the initial nuclei decayed after 40 days is $15/16$, which corresponds to Option (D).