it says that is wrong would the domain of the first one not be [-6,6]

You're correct in noting that the domain of the first component, \( 36 - t^2 \), is indeed restricted. Let's go through the calculation again with that in mind.

### Re-analysis:

1. **First component**: \( 36 - t^2 \)
   - This is defined for all real numbers \( t \), but it is also constrained because it represents the square root function. Since the square root function is not explicitly mentioned, but you indicated a possible restriction, let's assume you're referring to \( \sqrt{36 - t^2} \).
   - For \( 36 - t^2 \) to be non-negative (which would be required if this expression were under a square root), we need:
     \[
     36 - t^2 \geq 0
     \]
   - This inequality simplifies to:
     \[
     t^2 \leq 36
     \]
     \[
     -6 \leq t \leq 6
     \]
   - So, the domain of this component would be \( [-6, 6] \).

2. **Second component**: \( e^{-2t} \)
   - The exponential function \( e^{-2t} \) is defined for all real numbers \( t \).
   - Domain: \( (-\infty, \infty) \).

3. **Third component**: \( \ln(t + 4) \)
   - The natural logarithm \( \ln(t + 4) \) is defined for \( t + 4 > 0 \), which simplifies to \( t > -4 \).
   - Domain: \( (-4, \infty) \).

### Combined Domain:

Now, to find the overall domain of the vector function \( \mathbf{r}(t) = \left\langle 36 - t^2, e^{-2t}, \ln(t + 4) \right\rangle \):

- The domain of the first component is \( [-6, 6] \).
- The domain of the second component is \( (-\infty, \infty) \).
- The domain of the third component is \( (-4, \infty) \).

The overall domain is the intersection of these intervals. 

### Intersecting Domains:

- Intersection of \( [-6, 6] \) and \( (-\infty, \infty) \) is \( [-6, 6] \).
- Now, intersect \( [-6, 6] \) with \( (-4, \infty) \):
  - This intersection is \( [-4, 6] \).

### Final Answer:

The correct domain of the vector function \( \mathbf{r}(t) \) is:

\[ [-4, 6] \]

Would you like any further clarification or details on this?

Here are some additional questions to explore:

1. What is the range of the function \( e^{-2t} \) over the domain \( [-4, 6] \)?
2. How does the vector function behave at the endpoints \( t = -4 \) and \( t = 6 \)?
3. How would the domain change if the logarithmic term was \( \ln(t + 3) \)?
4. Can you find the range of the vector function \( \mathbf{r}(t) \)?
5. How would you graph the first component \( 36 - t^2 \) and indicate its domain?

**Tip:** Always remember to check each component individually and find the intersection of the domains when working with vector functions.

Explore the domain of the vector function r(t) = (36 - t^2, e^{-2t}, \ln(t + 4)) with a comprehensive analysis. Learn how to determine and intersect domains of exponential and logarithmic functions. Suitable for advanced high school students and beyond.

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