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The function [tex]D(t)[/tex] defines a traveler's distance from home, in miles, as a function of time, in hours.

[tex]\[
D(t) = \left\{
\begin{array}{ll}
300t + 125, & 0 \leq t \ \textless \ 2.5 \\
875, & 2.5 \leq t \leq 3.5 \\
75t + 612.5, & 3.5 \ \textless \ t \leq 6
\end{array}
\right.
\][/tex]

Which times and distances are represented by the function? Select three options.

A. The starting distance, at 0 hours, is 300 miles.
B. At 2 hours, the traveler is 725 miles from home.
C. At 2.5 hours, the traveler is still moving farther from home.
D. At 3 hours, the distance is constant, at 875 miles.
E. The total distance from home after 6 hours is [tex]1,062.5[/tex] miles.



Answer :

GinnyAnswer
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.

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