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**6. Halla la masa \( m \) de \( 10 \) gramos al segundo día y a los cinco días usando la ecuación \( m(t) = m_0 \cdot e^{-kt} \), donde \( m_0 = 10 \) gramos y \( k = 0.1 \text{ día}^{-1} \).**

### Step-by-Step Solution

We are given:
- Initial mass \( m_0 = 10 \) grams.
- Decay constant \( k = 0.1 \text{ día}^{-1} \).
- The equation \( m(t) = m_0 \cdot e^{-kt} \).

We need to find the mass \( m(t) \) at:
- \( t = 2 \) days.
- \( t = 5 \) days.

#### Step 1: Calculate the mass at \( t = 2 \) days.
Using the formula:
\[
m(2) = 10 \cdot e^{-0.1 \cdot 2}
\]
\[
m(2) = 10 \cdot e^{-0.2}
\]
\[
m(2) \approx 10 \cdot 0.8187 = 8.187 \text{ grams}
\]

#### Step 2: Calculate the mass at \( t = 5 \) days.
Using the formula:
\[
m(5) = 10 \cdot e^{-0.1 \cdot 5}
\]
\[
m(5) = 10 \cdot e^{-0.5}
\]
\[
m(5) \approx 10 \cdot 0.6065 = 6.065 \text{ grams}
\]

### Final Answers:
- The mass after 2 days is approximately \( 8.187 \) grams.
- The mass after 5 days is approximately \( 6.065 \) grams.

Would you like to go over more details or have any other questions? Here are five related questions that might interest you:

1. How do different values of \( k \) affect the decay rate in the equation \( m(t) = m_0 \cdot e^{-kt} \)?
2. What is the significance of the decay constant \( k \) in real-life applications?
3. How can this formula be used to model radioactive decay?
4. What would happen to the mass if \( k = 0 \)?
5. How would you calculate the time it takes for the mass to reduce to half its initial value?

**Tip:** When dealing with exponential decay, the decay constant \( k \) is crucial in determining how quickly the quantity decreases over time.

Learn how to calculate mass decay using the exponential function with a detailed step-by-step solution. This problem involves finding the mass at specific time points using the exponential decay formula. Suitable for high school students studying exponential functions.

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