To understand what [tex]\( t \)[/tex] represents in the given equation [tex]\( h(t) = -16t^2 + 32t + 10 \)[/tex], we need to analyze the components of the equation and their physical meanings in the context of the problem.
The equation [tex]\( h(t) = -16t^2 + 32t + 10 \)[/tex] models the height [tex]\( h \)[/tex] of the rocket at a time [tex]\( t \)[/tex].
1. Understanding the structure of the equation:
- The term [tex]\( -16t^2 \)[/tex]: This term represents the effect of gravity on the rocket's height. The coefficient [tex]\(-16\)[/tex] comes from the acceleration due to gravity, which is approximately [tex]\( -32 \)[/tex] feet per second squared, divided by 2 (hence, [tex]\( -16 \)[/tex]).
- The term [tex]\( 32t \)[/tex]: This represents the initial upward velocity of the rocket. The coefficient [tex]\( 32 \)[/tex] indicates the initial velocity in feet per second.
- The constant term [tex]\( 10 \)[/tex]: This represents the initial height of the rocket in feet when [tex]\( t = 0 \)[/tex].
2. Identifying what [tex]\( t \)[/tex] stands for:
- In the expression [tex]\( h(t) \)[/tex], [tex]\( t \)[/tex] is used to substitute different values to find the corresponding height of the rocket at different times.
- Each value of [tex]\( t \)[/tex] corresponds to a specific time after the rocket is released.
From this analysis, we can determine that [tex]\( t \)[/tex] is not representing the initial height of the rocket, the initial velocity of the rocket, nor the height after [tex]\( t \)[/tex] seconds on its own. Instead, [tex]\( t \)[/tex] represents the time variable in the equation. It specifically indicates:
[tex]\( \boxed{\text{the number of seconds after the rocket is released}} \)[/tex].
