To determine the domain of the function $h(t)$, which represents the height of a projectile as a function of time $t$, we need to consider the physical context of the problem.

The domain of $h(t)$ represents all the possible values of $t$ for which the height is defined and meaningful. Since $t$ represents time, it must be a non-negative value, as time cannot be negative in this context. Additionally, the time $t$ will only be valid from the moment the projectile is launched (at $t = 0$) until the moment it hits the ground.

Thus, the domain of $h(t)$ is:

$\text{Domain of } h(t): \, t \in [0, T]$

where $T$ is the time when the projectile hits the ground. The specific value of $T$ depends on the details of the function $h(t)$ (like initial velocity, height, and gravity), but generally, $T$ is the time when $h(t) = 0$ after being launched.

Would you like to go into further details about determining $T$ or have any other questions?

Here are 5 related questions you might consider:

1. How do you calculate the maximum height of the projectile?
2. What is the general form of the function $h(t)$ for a projectile?
3. How does the initial velocity affect the time $T$ when the projectile hits the ground?
4. What factors influence the shape of the $h(t)$ graph?
5. How do you find the time at which the projectile reaches its maximum height?

Tip: Understanding the domain in physical problems often involves considering the realistic constraints of the situation, such as non-negative time and conditions when the object is in motion.