Explore a vast range of topics and get informed answers at IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.
Sagot :
To determine the domain of the function [tex]\( H(t) \)[/tex], which represents the height of a model rocket as a function of time [tex]\( t \)[/tex], we need to understand the physical context of the problem.
1. Understanding the Function [tex]\( H(t) \)[/tex]:
- [tex]\( H(t) \)[/tex] denotes the height of the rocket at any given time [tex]\( t \)[/tex] and is typically measured from the moment of launch (which occurs at [tex]\( t = 0 \)[/tex]).
- The rocket will ascend to a certain maximum height, then descend back to the ground. This entire process is captured by the function [tex]\( H(t) \)[/tex].
2. Identifying the Practical Domain:
- The practical domain of [tex]\( H(t) \)[/tex] starts from the moment the rocket is launched, which corresponds to [tex]\( t = 0 \)[/tex].
- The rocket will continue to travel upwards until it reaches its peak, and then it will descend until it hits the ground. This means that [tex]\( t \)[/tex] can take on values from the launch time (0) up to the time it lands back on the ground.
3. Establishing the Time the Rocket Hits the Ground:
- Let's denote [tex]\( t_{max} \)[/tex] as the time when the rocket hits the ground. This is the point where the height [tex]\( H(t) \)[/tex] returns to zero.
- Therefore, the function [tex]\( H(t) \)[/tex] is defined for all [tex]\( t \)[/tex] starting at 0 and ending at [tex]\( t_{max} \)[/tex]. The exact value of [tex]\( t_{max} \)[/tex] depends on the specific dynamics of the rocket flight, which might not be provided explicitly.
4. Writing the Domain in Interval Notation:
- Since time [tex]\( t \)[/tex] cannot be negative in this context and [tex]\( t \)[/tex] ranges from the time of launch to the time the rocket lands, we can express the domain as:
[tex]\[ \text{The domain of } H(t) \text{ is } [0, t_{max}] \][/tex]
5. Conclusion:
- The domain of [tex]\( H(t) \)[/tex] is all non-negative real numbers starting from 0 and extending to the time when the rocket touches the ground again. Hence, the domain can be described as:
[tex]\[ \text{The domain of } H(t) \text{ is all non-negative real numbers, } [0, t_{max}] \][/tex]
This thorough analysis ensures we have correctly identified the domain for the height function [tex]\( H(t) \)[/tex] of the model rocket.
1. Understanding the Function [tex]\( H(t) \)[/tex]:
- [tex]\( H(t) \)[/tex] denotes the height of the rocket at any given time [tex]\( t \)[/tex] and is typically measured from the moment of launch (which occurs at [tex]\( t = 0 \)[/tex]).
- The rocket will ascend to a certain maximum height, then descend back to the ground. This entire process is captured by the function [tex]\( H(t) \)[/tex].
2. Identifying the Practical Domain:
- The practical domain of [tex]\( H(t) \)[/tex] starts from the moment the rocket is launched, which corresponds to [tex]\( t = 0 \)[/tex].
- The rocket will continue to travel upwards until it reaches its peak, and then it will descend until it hits the ground. This means that [tex]\( t \)[/tex] can take on values from the launch time (0) up to the time it lands back on the ground.
3. Establishing the Time the Rocket Hits the Ground:
- Let's denote [tex]\( t_{max} \)[/tex] as the time when the rocket hits the ground. This is the point where the height [tex]\( H(t) \)[/tex] returns to zero.
- Therefore, the function [tex]\( H(t) \)[/tex] is defined for all [tex]\( t \)[/tex] starting at 0 and ending at [tex]\( t_{max} \)[/tex]. The exact value of [tex]\( t_{max} \)[/tex] depends on the specific dynamics of the rocket flight, which might not be provided explicitly.
4. Writing the Domain in Interval Notation:
- Since time [tex]\( t \)[/tex] cannot be negative in this context and [tex]\( t \)[/tex] ranges from the time of launch to the time the rocket lands, we can express the domain as:
[tex]\[ \text{The domain of } H(t) \text{ is } [0, t_{max}] \][/tex]
5. Conclusion:
- The domain of [tex]\( H(t) \)[/tex] is all non-negative real numbers starting from 0 and extending to the time when the rocket touches the ground again. Hence, the domain can be described as:
[tex]\[ \text{The domain of } H(t) \text{ is all non-negative real numbers, } [0, t_{max}] \][/tex]
This thorough analysis ensures we have correctly identified the domain for the height function [tex]\( H(t) \)[/tex] of the model rocket.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
