Half-life of a radioactive substance is 2.9 days. Calculate the amount of 10 mg of substance left after 29 days.

Answer

To solve the problem of how much of a 10 mg radioactive substance is left after 29 days, given that its half-life is 2.9 days, we can follow these steps:

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

### Step 2: Calculate the number of half-lives in 29 days
To find out how many half-lives fit into 29 days, we divide the total time (29 days) by the half-life (2.9 days):

\[ \text{Number of half-lives} = \frac{29 \text{ days}}{2.9 \text{ days}} = 10 \]

### Step 3: Apply the half-life formula
The remaining quantity of the substance after a certain number of half-lives can be calculated using the formula:

\[ M_t = M_0 \left(\frac{1}{2}\right)^n \]

where:
- \(M_t\) is the remaining quantity after time \(t\),
- \(M_0\) is the initial quantity (10 mg in this case),
- \(n\) is the number of half-lives (10 in this case).

### Step 4: Substitute the values into the formula
Now we can substitute the values into the formula:

\[ M_t = 10 \text{ mg} \left(\frac{1}{2}\right)^{10} \]

### Step 5: Calculate \(\left(\frac{1}{2}\right)^{10}\)
Calculating \(\left(\frac{1}{2}\right)^{10}\):

\[ \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \]

### Step 6: Calculate the remaining quantity
Now we can calculate \(M_t\):

\[ M_t = 10 \text{ mg} \times \frac{1}{1024} = \frac{10}{1024} \text{ mg} \approx 0.00976 \text{ mg} \]

### Step 7: Final answer
Thus, the amount of the substance left after 29 days is approximately:

\[ M_t \approx 0.00976 \text{ mg} \]

### Summary
After 29 days, approximately 0.00976 mg of the radioactive substance remains.

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