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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]\[
D(t) =
\begin{cases}
300t + 125, & 0 \leq t \ \textless \ 2.5 \\
875, & 2.5 \leq t \leq 3.5 \\
75t + 612.5, & 3.5 \ \textless \ t \leq 6
\end{cases}
\][/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 1,062.5 miles.

Sagot :

GinnyAnswer
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.
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