Answer :
Let's analyze the function [tex]\( D(t) \)[/tex] and find the distances for the given times.
1. Starting Distance at 0 Hours:
- For [tex]\( t = 0 \)[/tex]:
[tex]\[ D(0) = 300 \cdot 0 + 125 = 125 \text{ miles} \][/tex]
Therefore, the starting distance at 0 hours is 125 miles, not 300 miles. So, the first option is incorrect.
2. Distance at 2 Hours:
- For [tex]\( t = 2 \)[/tex]:
[tex]\[ D(2) = 300 \cdot 2 + 125 = 600 + 125 = 725 \text{ miles} \][/tex]
Therefore, the traveler is 725 miles from home at 2 hours. The second option is correct.
3. At 2.5 Hours:
- For [tex]\( t = 2.5 \)[/tex]:
[tex]\[ D(2.5) = 875 \text{ miles} \][/tex]
The function reaches a constant value of 875 miles from 2.5 hours to 3.5 hours. Thus, at 2.5 hours, the traveler has stopped moving farther, so the third option is incorrect.
4. Distance at 3 Hours:
- For [tex]\( t = 3 \)[/tex]:
[tex]\[ D(3) = 875 \text{ miles} \][/tex]
The distance is constant at 875 miles from 2.5 hours to 3.5 hours. Therefore, at 3 hours, the distance is 875 miles. The fourth option is correct.
5. Total Distance After 6 Hours:
- For [tex]\( t = 6 \)[/tex]:
[tex]\[ D(6) = 75 \cdot 6 + 612.5 = 450 + 612.5 = 1062.5 \text{ miles} \][/tex]
Therefore, the total distance from home after 6 hours is 1062.5 miles. The fifth option is correct.
So, the correct options based on the analysis are:
1. At 2 hours, the traveler is 725 miles from home.
2. At 3 hours, the distance is constant, at 875 miles.
3. The total distance from home after 6 hours is 1062.5 miles.
1. Starting Distance at 0 Hours:
- For [tex]\( t = 0 \)[/tex]:
[tex]\[ D(0) = 300 \cdot 0 + 125 = 125 \text{ miles} \][/tex]
Therefore, the starting distance at 0 hours is 125 miles, not 300 miles. So, the first option is incorrect.
2. Distance at 2 Hours:
- For [tex]\( t = 2 \)[/tex]:
[tex]\[ D(2) = 300 \cdot 2 + 125 = 600 + 125 = 725 \text{ miles} \][/tex]
Therefore, the traveler is 725 miles from home at 2 hours. The second option is correct.
3. At 2.5 Hours:
- For [tex]\( t = 2.5 \)[/tex]:
[tex]\[ D(2.5) = 875 \text{ miles} \][/tex]
The function reaches a constant value of 875 miles from 2.5 hours to 3.5 hours. Thus, at 2.5 hours, the traveler has stopped moving farther, so the third option is incorrect.
4. Distance at 3 Hours:
- For [tex]\( t = 3 \)[/tex]:
[tex]\[ D(3) = 875 \text{ miles} \][/tex]
The distance is constant at 875 miles from 2.5 hours to 3.5 hours. Therefore, at 3 hours, the distance is 875 miles. The fourth option is correct.
5. Total Distance After 6 Hours:
- For [tex]\( t = 6 \)[/tex]:
[tex]\[ D(6) = 75 \cdot 6 + 612.5 = 450 + 612.5 = 1062.5 \text{ miles} \][/tex]
Therefore, the total distance from home after 6 hours is 1062.5 miles. The fifth option is correct.
So, the correct options based on the analysis are:
1. At 2 hours, the traveler is 725 miles from home.
2. At 3 hours, the distance is constant, at 875 miles.
3. The total distance from home after 6 hours is 1062.5 miles.
