While on Casey's computer, you notice that a model for a recent experiment is completed. The mo gives the temperature $$ (T) $$ as a function of time $$ (t) $$ as given below. As before, a time of $$ t=0 $$ represe the time of data collection, with negative values representing earlier times. Knowing Casey, they want to know the domain, range, and other information about the model. You review the model provide the information that Casey will request. $$ T(t)=\ln (6 t+36)+1.6 $$ Enter the domain in interval notation. To enter $$ \infty $$, type infinity. The vertical asymptote is $$ t=\square $$ As $$ t $$ approaches the vertical asymptote, $$ \quad T(t) \rightarrow \square $$ As $$ t $$ approaches $$ \infty $$,

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To analyze the function $$ T(t) = \ln(6t + 36) + 1.6 $$, we need to determine the domain, vertical asymptote, and behavior of the function as $$ t $$ approaches certain values.

### Domain
The logarithmic function $$ \ln(x) $$ is defined only for $$ x > 0 $$. Therefore, we need to find when the argument of the logarithm is positive:

$$
6t + 36 > 0
$$

Solving for $$ t $$:

$$
6t > -36 \
t > -6
$$

Thus, the domain of $$ T(t) $$ in interval notation is:

$$
(-6, \infty)
$$

### Vertical Asymptote
The vertical asymptote occurs where the function approaches negative infinity, which happens when the argument of the logarithm approaches zero:

$$
6t + 36 = 0 \
6t = -36 \
t = -6
$$

So, the vertical asymptote is:

$$
t = -6
$$

### Behavior Near the Vertical Asymptote
As $$ t $$ approaches the vertical asymptote from the right (i.e., $$ t \to -6^+ $$), the argument of the logarithm approaches zero:

$$
T(t) = \ln(6t + 36) + 1.6 \to \ln(0) + 1.6 \to -\infty + 1.6 = -\infty
$$

Thus, as $$ t $$ approaches the vertical asymptote, we have:

$$
T(t) \rightarrow -\infty
$$

### Behavior as $$ t $$ Approaches Infinity
As $$ t $$ approaches infinity, the term $$ 6t + 36 $$ also approaches infinity:

$$
T(t) = \ln(6t + 36) + 1.6 \to \ln(\infty) + 1.6 = \infty + 1.6 = \infty
$$

Thus, as $$ t $$ approaches infinity, we have:

$$
T(t) \rightarrow \infty
$$

### Summary
- **Domain**: $$(-6, \infty)$$
- **Vertical asymptote**: $$t = -6$$
- **As $$ t $$ approaches the vertical asymptote**: $$T(t) \rightarrow -\infty$$
- **As $$ t $$ approaches infinity**: $$T(t) \rightarrow \infty$$
- Domain: $$(-6, \infty)$$ 	- Vertical asymptote: $$t = -6$$ 	- As $$ t $$ approaches the vertical asymptote: $$T(t) \rightarrow -\infty$$ 	- As $$ t $$ approaches infinity: $$T(t) \rightarrow \infty$$

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Domain

Vertical Asymptote

Behavior Near the Vertical Asymptote

Behavior as Approaches Infinity

Summary

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