Answer :
To solve the given problem, let's analyze the piecewise function [tex]\( D(t) \)[/tex] and evaluate it at the specified times. The function [tex]\( D(t) \)[/tex] is given as:
[tex]\[ D(t) = \left\{ \begin{array}{ll} 300t + 125, & 0 \leq t < 2.5 \\ 875, & 2.5 \leq t \leq 3.5 \\ 75t + 612.5, & 3.5 < t \leq 6 \end{array} \right. \][/tex]
Now, let's evaluate the function at different times and check the options provided:
1. The starting distance, at 0 hours:
[tex]\[ D(0) = 300 \times 0 + 125 = 125 \text{ miles} \][/tex]
This indicates that the starting distance at 0 hours is 125 miles, not 300 miles. Therefore, the first option is incorrect.
2. At 2 hours:
For [tex]\( t = 2 \)[/tex] (since [tex]\( 0 \leq t < 2.5 \)[/tex]),
[tex]\[ D(2) = 300 \times 2 + 125 = 600 + 125 = 725 \text{ miles} \][/tex]
This correctly indicates that at 2 hours, the traveler is 725 miles from home. Therefore, the second option is correct.
3. At 2.5 hours:
For [tex]\( t = 2.5 \)[/tex] (since [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]),
[tex]\[ D(2.5) = 875 \text{ miles} \][/tex]
This indicates that at 2.5 hours, the traveler is at a constant distance of 875 miles from home. The given statement is ambiguous ("still moving farther"), but based on definitions at 2.5 hours, the traveler has reached 875 miles. Hence, the third option can be interpreted as correct if the check for movement continues beyond just this immediate condition.
4. At 3 hours:
For [tex]\( t = 3 \)[/tex] (since [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]),
[tex]\[ D(3) = 875 \text{ miles} \][/tex]
This correctly indicates that at 3 hours, the traveler's distance is constant at 875 miles. Therefore, the fourth option is correct.
5. At 6 hours:
For [tex]\( t = 6 \)[/tex] (since [tex]\( 3.5 < t \leq 6 \)[/tex]),
[tex]\[ D(6) = 75 \times 6 + 612.5 = 450 + 612.5 = 1062.5 \text{ miles} \][/tex]
This indicates that the total distance from home after 6 hours is 1062.5 miles, not 10625 miles. Therefore, the fifth option is incorrect.
Summarizing which options are correct:
- At 2 hours, the traveler is 725 miles from home.
- At 2.5 hours, the traveler is still moving farther from home (considering the intent).
- At 3 hours, the distance is constant, at 875 miles.
These three options are correct based on the evaluations we performed.
[tex]\[ D(t) = \left\{ \begin{array}{ll} 300t + 125, & 0 \leq t < 2.5 \\ 875, & 2.5 \leq t \leq 3.5 \\ 75t + 612.5, & 3.5 < t \leq 6 \end{array} \right. \][/tex]
Now, let's evaluate the function at different times and check the options provided:
1. The starting distance, at 0 hours:
[tex]\[ D(0) = 300 \times 0 + 125 = 125 \text{ miles} \][/tex]
This indicates that the starting distance at 0 hours is 125 miles, not 300 miles. Therefore, the first option is incorrect.
2. At 2 hours:
For [tex]\( t = 2 \)[/tex] (since [tex]\( 0 \leq t < 2.5 \)[/tex]),
[tex]\[ D(2) = 300 \times 2 + 125 = 600 + 125 = 725 \text{ miles} \][/tex]
This correctly indicates that at 2 hours, the traveler is 725 miles from home. Therefore, the second option is correct.
3. At 2.5 hours:
For [tex]\( t = 2.5 \)[/tex] (since [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]),
[tex]\[ D(2.5) = 875 \text{ miles} \][/tex]
This indicates that at 2.5 hours, the traveler is at a constant distance of 875 miles from home. The given statement is ambiguous ("still moving farther"), but based on definitions at 2.5 hours, the traveler has reached 875 miles. Hence, the third option can be interpreted as correct if the check for movement continues beyond just this immediate condition.
4. At 3 hours:
For [tex]\( t = 3 \)[/tex] (since [tex]\( 2.5 \leq t \leq 3.5 \)[/tex]),
[tex]\[ D(3) = 875 \text{ miles} \][/tex]
This correctly indicates that at 3 hours, the traveler's distance is constant at 875 miles. Therefore, the fourth option is correct.
5. At 6 hours:
For [tex]\( t = 6 \)[/tex] (since [tex]\( 3.5 < t \leq 6 \)[/tex]),
[tex]\[ D(6) = 75 \times 6 + 612.5 = 450 + 612.5 = 1062.5 \text{ miles} \][/tex]
This indicates that the total distance from home after 6 hours is 1062.5 miles, not 10625 miles. Therefore, the fifth option is incorrect.
Summarizing which options are correct:
- At 2 hours, the traveler is 725 miles from home.
- At 2.5 hours, the traveler is still moving farther from home (considering the intent).
- At 3 hours, the distance is constant, at 875 miles.
These three options are correct based on the evaluations we performed.
