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Answer :
Sure, let's tackle the problem step-by-step.
### Problem:
We have the set of ordered pairs (t, d) where [tex]\( t \)[/tex] is the time in seconds after someone jumps out of a plane at 4000 meters to go skydiving, and [tex]\( d \)[/tex] is the displacement above the ground. We need to find the domain and range for this situation.
### Solution:
1. Understanding the Scenario:
- The person starts at an altitude of 4000 meters.
- As time [tex]\( t \)[/tex] progresses, they will fall toward the ground (0 meters).
2. Determine the Domain:
- The domain is the set of all possible values for [tex]\( t \)[/tex], which is the time from when the person jumps out of the plane until they reach the ground.
- Initially, [tex]\( t = 0 \)[/tex] (jumping moment).
- They will reach the ground after a certain period. Let's denote this total time by [tex]\( T \)[/tex].
3. Determine the Range:
- The range is the set of all possible values for [tex]\( d \)[/tex], which is the displacement above the ground.
- Initially, [tex]\( d = 4000 \)[/tex] meters (starting altitude).
- Finally, [tex]\( d = 0 \)[/tex] meters (ground level).
4. Calculate Total Time in Freefall and Parachute Descent:
- Freefall Phase:
- Let's assume the person free-falls until reaching 1000 meters above the ground.
- The time for free-fall under gravity [tex]\( g \)[/tex] (approximated as 9.8 m/s²) can be derived from the formula for displacement in free-fall: [tex]\( d = \frac{1}{2}gt^2 \)[/tex].
- So, the time to fall from 4000 meters to 1000 meters is found using: [tex]\( t_{\text{freefall}} = \sqrt{\frac{2d}{g}} \)[/tex], where [tex]\( d = 3000 \)[/tex] meters.
- The time for freefall [tex]\( t_{\text{freefall}} \approx 14.29 \)[/tex] seconds.
- Parachute-Descent Phase:
- After the parachute opens at 1000 meters, the descent rate is much slower. Let's assume it is a constant rate of 5 m/s.
- The time to descend the remaining 1000 meters would then be: [tex]\( t_{\text{parachute}} = \frac{1000}{5} \)[/tex].
- The time for parachute descent [tex]\( t_{\text{parachute}} \approx 200 \)[/tex] seconds.
5. Calculate Total Time:
- The total time, [tex]\( T \)[/tex], is the sum of the free-fall time and the parachute-descent time.
- [tex]\( T = t_{\text{freefall}} + t_{\text{parachute}} \approx 14.29 + 200 = 214.29 \)[/tex] seconds.
### Conclusion:
- Domain:
- The domain of [tex]\( t \)[/tex] is from 0 to the total time, which is 0 to [tex]\( 214.29 \)[/tex] seconds.
- So, the domain is: [tex]\([0, 214.29)\)[/tex] seconds.
- Range:
- The range of [tex]\( d \)[/tex] is from the initial altitude to the ground level.
- So, the range is: [tex]\([0, 4000]\)[/tex] meters.
To summarize:
- The domain is [tex]\([0, 214.29)\)[/tex] seconds.
- The range is [tex]\([0, 4000]\)[/tex] meters.
### Problem:
We have the set of ordered pairs (t, d) where [tex]\( t \)[/tex] is the time in seconds after someone jumps out of a plane at 4000 meters to go skydiving, and [tex]\( d \)[/tex] is the displacement above the ground. We need to find the domain and range for this situation.
### Solution:
1. Understanding the Scenario:
- The person starts at an altitude of 4000 meters.
- As time [tex]\( t \)[/tex] progresses, they will fall toward the ground (0 meters).
2. Determine the Domain:
- The domain is the set of all possible values for [tex]\( t \)[/tex], which is the time from when the person jumps out of the plane until they reach the ground.
- Initially, [tex]\( t = 0 \)[/tex] (jumping moment).
- They will reach the ground after a certain period. Let's denote this total time by [tex]\( T \)[/tex].
3. Determine the Range:
- The range is the set of all possible values for [tex]\( d \)[/tex], which is the displacement above the ground.
- Initially, [tex]\( d = 4000 \)[/tex] meters (starting altitude).
- Finally, [tex]\( d = 0 \)[/tex] meters (ground level).
4. Calculate Total Time in Freefall and Parachute Descent:
- Freefall Phase:
- Let's assume the person free-falls until reaching 1000 meters above the ground.
- The time for free-fall under gravity [tex]\( g \)[/tex] (approximated as 9.8 m/s²) can be derived from the formula for displacement in free-fall: [tex]\( d = \frac{1}{2}gt^2 \)[/tex].
- So, the time to fall from 4000 meters to 1000 meters is found using: [tex]\( t_{\text{freefall}} = \sqrt{\frac{2d}{g}} \)[/tex], where [tex]\( d = 3000 \)[/tex] meters.
- The time for freefall [tex]\( t_{\text{freefall}} \approx 14.29 \)[/tex] seconds.
- Parachute-Descent Phase:
- After the parachute opens at 1000 meters, the descent rate is much slower. Let's assume it is a constant rate of 5 m/s.
- The time to descend the remaining 1000 meters would then be: [tex]\( t_{\text{parachute}} = \frac{1000}{5} \)[/tex].
- The time for parachute descent [tex]\( t_{\text{parachute}} \approx 200 \)[/tex] seconds.
5. Calculate Total Time:
- The total time, [tex]\( T \)[/tex], is the sum of the free-fall time and the parachute-descent time.
- [tex]\( T = t_{\text{freefall}} + t_{\text{parachute}} \approx 14.29 + 200 = 214.29 \)[/tex] seconds.
### Conclusion:
- Domain:
- The domain of [tex]\( t \)[/tex] is from 0 to the total time, which is 0 to [tex]\( 214.29 \)[/tex] seconds.
- So, the domain is: [tex]\([0, 214.29)\)[/tex] seconds.
- Range:
- The range of [tex]\( d \)[/tex] is from the initial altitude to the ground level.
- So, the range is: [tex]\([0, 4000]\)[/tex] meters.
To summarize:
- The domain is [tex]\([0, 214.29)\)[/tex] seconds.
- The range is [tex]\([0, 4000]\)[/tex] meters.
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