To model the height $h$ of the object $t$ seconds after it was released, you can use the following quadratic function derived from the principles of projectile motion:

$h(t) = -16t^2 + v_0 t + h_0$

Where:

• $h(t)$ is the height in feet at time $t$.
• $v_0$ is the initial upward velocity (30 ft/s).
• $h_0$ is the initial height (12 ft).
• The term $-16t^2$ accounts for the effect of gravity (approximately $32 \, \text{ft/s}^2$, hence divided by 2).

Substituting the values:

$h(t) = -16t^2 + 30t + 12$

This function gives the height of the object at any time $t$ after it has been released.

Would you like further details or have any questions? Here are some related questions for you to consider:

1. What is the maximum height the object reaches?
2. How long does it take for the object to hit the ground?
3. What is the height of the object after 1 second?
4. How does changing the initial velocity affect the height function?
5. What would happen if the object was thrown downward instead of upward?

Tip: When modeling projectile motion, always remember to account for both the initial height and velocity in your function.