To analyze the given problem, let's break down the components of the function [tex]\( h(t) = -16t^2 + 32t + 10 \)[/tex], which models the height of a rocket at any time [tex]\( t \)[/tex] seconds after it is released.
1. The term [tex]\( t \)[/tex] in [tex]\( h(t) \)[/tex]:
- In the context of the problem, [tex]\( t \)[/tex] represents the number of seconds that have passed since the rocket was released.
2. The constant term (10) in the equation [tex]\( h(t) \)[/tex]:
- The constant term in the height function equation is 10. This term represents the initial height of the rocket at [tex]\( t = 0 \)[/tex]. Therefore, [tex]\( 10 \)[/tex] is the initial height of the rocket in feet.
3. The coefficient of the linear term (32) in the equation [tex]\( h(t) \)[/tex]:
- The coefficient of the [tex]\( t \)[/tex] term (linear term) in the equation is 32. This term represents the initial velocity of the rocket. Thus, [tex]\( 32 \)[/tex] is the initial velocity of the rocket in feet per second (ft/s).
4. The function [tex]\( h(t) \)[/tex] itself:
- The whole function [tex]\( h(t) \)[/tex] represents the height of the rocket after [tex]\( t \)[/tex] seconds. Therefore, for any given value of [tex]\( t \)[/tex], [tex]\( h(t) \)[/tex] gives the height of the rocket in feet.
To summarize:
- "The number of seconds after the rocket is released" refers to [tex]\( t \)[/tex].
- "The initial height of the rocket" is represented by the constant term 10.
- "The initial velocity of the rocket" is the coefficient 32.
- "The height of the rocket after [tex]\( t \)[/tex] seconds" is represented by [tex]\( h(t) \)[/tex], the entire function.
Thus, the detailed breakdown is as follows:
- [tex]\( t \)[/tex] represents the number of seconds after the rocket is released.
- 10 is the initial height of the rocket.
- 32 is the initial velocity of the rocket.
- [tex]\( h(t) \)[/tex] is the height of the rocket after [tex]\( t \)[/tex] seconds.
The answer to the respective parts of the question are:
1. The number of seconds after the rocket is released: [tex]\( t \)[/tex]
2. The initial height of the rocket: 10 feet
3. The initial velocity of the rocket: 32 ft/s
4. The height of the rocket after [tex]\( t \)[/tex] seconds: [tex]\( h(t) \)[/tex]
