To model the height \( h(t) \) of a ball dropped from a 30-foot tall building after \( t \) seconds, we can use the equation of motion under the influence of gravity. When an object is dropped (with no initial velocity) from a certain height, its height at time \( t \) can be represented by the following equation:
\[ h(t) = h_0 - \frac{1}{2} g t^2 \]
Where:
- \( h_0 \) is the initial height (in feet), which is 30 feet in this case.
- \( g \) is the acceleration due to gravity, approximately \( 32 \) feet per second squared.
- \( t \) is the time in seconds.
Substituting the values into the equation, we get:
\[ h(t) = 30 - \frac{1}{2} \cdot 32 \cdot t^2 \]
This simplifies to:
\[ h(t) = 30 - 16t^2 \]
So, the model representing the height of the ball from the ground \( t \) seconds after it is dropped is:
\[ h(t) = 30 - 16t^2 \]
This equation will describe the height of the ball as a function of time until it hits the ground (when \( h(t) = 0 \)).