Let's evaluate the function [tex]\( D(t) \)[/tex] using the provided points in time to identify which times and distances are accurately represented:
1. Distance at 0 hours:
[tex]\[
D(0) = 300 \cdot 0 + 125 = 125 \ \text{miles}
\][/tex]
So, the traveler starts at a distance of 125 miles from home.
2. Distance at 2 hours:
[tex]\[
D(2) = 300 \cdot 2 + 125 = 600 + 125 = 725 \ \text{miles}
\][/tex]
At 2 hours, the traveler is 725 miles from home.
3. Distance at 2.5 hours:
[tex]\[
D(2.5) = 875 \ \text{miles}
\][/tex]
At 2.5 hours, the distance is constant at 875 miles indicating the traveler is not moving farther but has stopped.
4. Distance at 3 hours:
[tex]\[
D(3) = 875 \ \text{miles}
\][/tex]
At 3 hours, the distance remains constant at 875 miles showing that the traveler is still not moving.
5. Distance at 6 hours:
[tex]\[
D(6) = 75 \cdot 6 + 612.5 = 450 + 612.5 = 1062.5 \ \text{miles}
\][/tex]
After 6 hours, the traveler is 1062.5 miles from home.
The correct statements based on our evaluations are:
- At 2 hours, the traveler is 725 miles from home.
- At 3 hours, the distance is constant at 875 miles.
- The total distance from home after 6 hours is 1062.5 miles.
