To determine what \( t \) represents in the context of the provided equation \( h(t) = -16t^2 + 32t + 10 \), let's break down the components of this quadratic equation.
1. Understanding the Equation:
- \( h(t) \) is the height of the rocket after \( t \) seconds.
- The variable \( t \) typically represents time in seconds in such equations.
- The constants and coefficients in the equation help define the path of the rocket over time.
2. Analyzing Each Term:
- The term \( -16t^2 \) is associated with the effect of gravity. It shows how the height changes due to gravity's decelerative effect on the rocket's vertical motion over time.
- The term \( 32t \) represents the contribution to the height due to the initial velocity of the rocket.
- The constant term \( 10 \) represents the initial height from which the rocket was released.
3. Interpreting \( t \):
- Given that \( h(t) = -16t^2 + 32t + 10 \) models the height of the rocket, and we typically plot height as a function of time, \( t \) must represent time in seconds.
- Therefore, \( t \) indicates the number of seconds after the rocket is released.
4. Matching \( t \) to Given Options:
- Given the choices:
1. The number of seconds after the rocket is released
2. The initial height of the rocket
3. The initial velocity of the rocket
4. The height of the rocket after \( t \) seconds
- We see that option 1 correctly identifies \( t \) as the number of seconds after the rocket is released.
In conclusion, \( t \) represents the number of seconds after the rocket is released. Therefore, the correct option is:
1. The number of seconds after the rocket is released.
