Let's analyze the function [tex]\( D(t) \)[/tex] and calculate the distances at specific times using the piecewise definition provided:
1. Starting distance, at [tex]\( t = 0 \)[/tex] hours:
For [tex]\( 0 \leq t < 2.5 \)[/tex]:
[tex]\[
D(0) = 300 \cdot 0 + 125 = 125 \text{ miles}
\][/tex]
Therefore, the starting distance is 125 miles, not 300 miles. The given statement "The starting distance, at 0 hours, is 300 miles" is false.
2. At [tex]\( t = 2 \)[/tex] hours:
For [tex]\( 0 \leq t < 2.5 \)[/tex]:
[tex]\[
D(2) = 300 \cdot 2 + 125 = 600 + 125 = 725 \text{ miles}
\][/tex]
Therefore, at 2 hours, the traveler is indeed 725 miles from home. The statement "At 2 hours, the traveler is 725 miles from home" is true.
3. At [tex]\( t = 2.5 \)[/tex] hours:
For [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]:
[tex]\[
D(2.5) = 875 \text{ miles}
\][/tex]
Therefore, at 2.5 hours, the distance is constant, at 875 miles. The statement "At 2.5 hours, the traveler is still moving farther from home" is false, as the traveler is actually at a constant distance of 875 miles during this interval.
4. At [tex]\( t = 3 \)[/tex] hours:
For [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]:
[tex]\[
D(3) = 875 \text{ miles}
\][/tex]
Therefore, at 3 hours, the distance is constant at 875 miles. The statement "At 3 hours, the distance is constant, at 875 miles" is true.
5. Total distance from home after [tex]\( t = 6 \)[/tex] hours:
For [tex]\( 3.5 < t \leq 6 \)[/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 [tex]\(1062.5\)[/tex] miles. The statement "The total distance from home after 6 hours is [tex]\(1062.5\)[/tex] miles" is true.
Summing up:
- "At 2 hours, the traveler is 725 miles from home." is true.
- "At 3 hours, the distance is constant, at 875 miles." is true.
- "The total distance from home after 6 hours is [tex]\( 1062.5 \)[/tex] miles." is true.
