To solve the question about what the variable [tex]\( t \)[/tex] represents in the given height equation [tex]\( h(t) = -16t^2 + 32t + 10 \)[/tex], let's break it down step-by-step.
1. Understand the Equation: The equation [tex]\( h(t) = -16t^2 + 32t + 10 \)[/tex] is a quadratic equation that describes the height of the rocket as a function of time [tex]\( t \)[/tex]. Here, [tex]\( h(t) \)[/tex] represents the height of the rocket at a specific time [tex]\( t \)[/tex].
2. Identify Parts of the Quadratic Equation:
- The term [tex]\( -16t^2 \)[/tex] indicates how the height changes with time due to the acceleration of gravity (assuming it's modeled in feet per second squared).
- The term [tex]\( 32t \)[/tex] represents the initial velocity effect on the height.
- The constant term [tex]\( 10 \)[/tex] represents the initial height of the rocket at the moment of its release.
3. What Does [tex]\( t \)[/tex] Represent?
- In the context of this equation, [tex]\( t \)[/tex] is placed in a position where it multiplies with time-dependent factors and affects the changing height of the rocket.
- Therefore, [tex]\( t \)[/tex] directly influences the height [tex]\( h(t) \)[/tex] based on how much time has passed since the rocket was released.
Given this understanding and analyzing the options:
- Option (a): [tex]\( t \)[/tex] is "the number of seconds after the rocket is released" aligns perfectly with the role of [tex]\( t \)[/tex] in the equation. It signifies the elapsed time affecting the calculated height at any given moment.
- Option (b): "the initial height of the rocket" is incorrect because the initial height is represented by the constant term [tex]\( 10 \)[/tex] when [tex]\( t = 0 \)[/tex].
- Option (c): "the initial velocity of the rocket" is incorrect since the term representing the initial velocity effect is [tex]\( 32t \)[/tex] and [tex]\( t \)[/tex] itself is not the initial velocity.
- Option (d): "the height of the rocket after [tex]\( t \)[/tex] seconds" is incorrect because [tex]\( h(t) \)[/tex] represents the height, not [tex]\( t \)[/tex].
Based on the analysis, the correct interpretation is:
[tex]\( t \)[/tex] represents the number of seconds after the rocket is released.
