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This problem deals with the exponential decay of a radioactive isotope, where the decay rate is proportional to the amount present. The equation used to describe this behavior is given as:

\[
\frac{dQ}{dt} = rQ
\]

Where:
- \( Q(t) \) is the amount of radioactive isotope at time \( t \),
- \( r \) is the decay constant, and
- \( t \) is the time (in this case, one week).

The process to find the decay constant \( r \) can be described as follows:

### Step-by-step solution:

1. **Set up the decay formula:**

   The general solution for the differential equation \( \frac{dQ}{dt} = rQ \) is an exponential function:

   \[
   Q(t) = Q_0 e^{rt}
   \]

   Where:
   - \( Q_0 \) is the initial quantity,
   - \( Q(t) \) is the quantity at time \( t \),
   - \( r \) is the decay constant, and
   - \( t \) is the time.

2. **Apply the initial conditions:**

   From the problem, we are given:
   - \( Q_0 = 300 \, \text{mg} \) (the initial amount),
   - \( Q(1) = 247.2 \, \text{mg} \) (the amount after 1 week, \( t = 1 \)).

   Substituting these into the equation \( Q(t) = Q_0 e^{rt} \):

   \[
   247.2 = 300 e^{r(1)}
   \]

3. **Solve for the decay constant \( r \):**

   Divide both sides by 300:

   \[
   \frac{247.2}{300} = e^{r}
   \]

   Simplifying:

   \[
   0.824 = e^{r}
   \]

   Take the natural logarithm on both sides:

   \[
   \ln(0.824) = r
   \]

   Using the value of \( \ln(0.824) \):

   \[
   r \approx -0.193
   \]

   So the decay constant \( r \) is approximately \( -0.193 \, \text{week}^{-1} \).

### Conclusion:

The decay constant \( r \approx -0.193 \, \text{week}^{-1} \). The negative sign indicates a decay process, but often in problems, the magnitude of the decay constant is reported, which would be \( 0.193 \, \text{week}^{-1} \).

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Let me know if you need further details or if you have any questions!

### Here are five related questions:
1. How can we calculate the half-life of the isotope using the decay constant?
2. What is the remaining quantity of the isotope after two weeks?
3. How would the decay constant change if the initial quantity was different?
4. Can we apply the same method for different decay rates in different time units (e.g., months or years)?
5. How does the exponential decay model relate to real-world applications in medicine and archaeology?

### Tip:
In exponential decay problems, remember that the decay constant \( r \) determines how fast the quantity reduces over time, and it is directly related to the half-life of the substance.

Un isótopo radioactivo se desintegra a una rapidez proporcional a la cantidad que hay presente. Si se reducen de 300 miligramos de este elemento a 247.2 miligramos en una semana. ¿Cuál de las siguientes opciones se aproxima más a la constante de decaimiento del isótopo?

Aprende cómo calcular la constante de decaimiento de un isótopo radioactivo que se reduce de 300 mg a 247.2 mg en una semana. Este problema utiliza ecuaciones diferenciales y funciones exponenciales, ideales para estudiantes de nivel avanzado.

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