Samples of two radioactive nuclides, `X` and `Y`, each have equal activity `A` at time `t=0`. X has a half-life of `24 years` and `Y` a half-life of `16 years`. The samples are mixed together. What will be the total activity of the mixture at `t=48` years?

Options

A. `(1)/(2) A_(0)`
B. `(1)/(4) A_(0)`
C. `(3)/(16) A_(0)`
D. `(3)/(8) A_(0)`

Explanation

To solve the problem, we need to calculate the remaining activity of each radioactive nuclide after 48 years and then sum them up to get the total activity of the mixture.

### Step 1: Understand the half-life concept
The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. After each half-life, the remaining activity is halved.

### Step 2: Determine the number of half-lives for each nuclide
- For nuclide X, the half-life is 24 years.
- For nuclide Y, the half-life is 16 years.
- We need to find out how many half-lives fit into 48 years.

For X:
\[ \text{Number of half-lives for X} = \frac{48 \text{ years}}{24 \text{ years/half-life}} = 2 \text{ half-lives} \]

For Y:
\[ \text{Number of half-lives for Y} = \frac{48 \text{ years}}{16 \text{ years/half-life}} = 3 \text{ half-lives} \]

### Step 3: Calculate the remaining activity for each nuclide
Let the initial activity of both nuclides at \( t = 0 \) be \( A_0 \).

For nuclide X after 2 half-lives:
\[ A_X = \frac{A_0}{2^2} = \frac{A_0}{4} \]

For nuclide Y after 3 half-lives:
\[ A_Y = \frac{A_0}{2^3} = \frac{A_0}{8} \]

### Step 4: Calculate the total activity of the mixture
Now, we can find the total activity of the mixture:
\[ A_{\text{total}} = A_X + A_Y = \frac{A_0}{4} + \frac{A_0}{8} \]

To add these fractions, we need a common denominator:
\[ A_{\text{total}} = \frac{2A_0}{8} + \frac{A_0}{8} = \frac{3A_0}{8} \]

### Final Answer
The total activity of the mixture at \( t = 48 \) years is:
\[ \boxed{\frac{3A_0}{8}} \]