A radioactive substance has decay constant of 0.231 `"day"^(-1)`. Calculate the percentage of the substance left after 6 days.

Answer

To solve the problem of calculating the percentage of a radioactive substance left after 6 days given a decay constant of 0.231 day^(-1), we can follow these steps:

### Step 1: Understand the formula
The mass of a radioactive substance remaining after time \( t \) can be expressed using the formula:
\[ M(t) = M_0 \cdot e^{-\lambda t} \]
where:
- \( M(t) \) is the mass remaining after time \( t \),
- \( M_0 \) is the initial mass,
- \( \lambda \) is the decay constant,
- \( t \) is the time elapsed.

### Step 2: Substitute the known values
In this case, we know:
- \( \lambda = 0.231 \, \text{day}^{-1} \)
- \( t = 6 \, \text{days} \)

Substituting these values into the formula gives:
\[ M(6) = M_0 \cdot e^{-0.231 \times 6} \]

### Step 3: Calculate the exponent
Now we need to calculate the exponent:
\[ -0.231 \times 6 = -1.386 \]

### Step 4: Calculate \( e^{-1.386} \)
Next, we calculate \( e^{-1.386} \):
\[ e^{-1.386} \approx 0.250 \]

### Step 5: Find the remaining mass
Now we can express the remaining mass:
\[ M(6) = M_0 \cdot 0.250 \]

### Step 6: Calculate the percentage of the substance left
To find the percentage of the substance left, we can use the formula:
\[ \text{Percentage remaining} = \left( \frac{M(6)}{M_0} \right) \times 100 \]
Substituting \( M(6) \):
\[ \text{Percentage remaining} = \left( 0.250 \right) \times 100 = 25\% \]

### Final Answer
The percentage of the substance left after 6 days is **25%**.

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