Answered

Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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 described by the function? Select three options.

A. The starting distance, at [tex]\( t = 0 \)[/tex].

B. At 2 hours, the traveler is _______ miles from home.

C. At 2.5 hours, the traveler is _______ miles from home.

D. At 3 hours, the traveler is _______ miles from home.

E. The total distance traveled is _______ miles.

Sagot :

GinnyAnswer
To solve this problem, we need to evaluate the piecewise function [tex]\( D(t) \)[/tex] at the specified times to determine the traveler's distance from home. Here's how we can do it step-by-step:

### 1. The Starting Distance, at [tex]\( t = 0 \)[/tex]:

For [tex]\( t \)[/tex] in the interval [tex]\( 0 \leq t < 2.5 \)[/tex], the function [tex]\( D(t) \)[/tex] is given by [tex]\( D(t) = 300t + 125 \)[/tex].
- When [tex]\( t = 0 \)[/tex]:

[tex]\[ D(0) = 300(0) + 125 = 125 \][/tex]

So, the starting distance is [tex]\( 125 \)[/tex] miles.

### 2. At 2 Hours, the Traveler Is:

For [tex]\( t \)[/tex] in the interval [tex]\( 0 \leq t < 2.5 \)[/tex], the function [tex]\( D(t) \)[/tex] is given by [tex]\( D(t) = 300t + 125 \)[/tex].
- When [tex]\( t = 2 \)[/tex]:

[tex]\[ D(2) = 300(2) + 125 = 600 + 125 = 725 \][/tex]

So, at 2 hours, the traveler is [tex]\( 725 \)[/tex] miles from home.

### 3. At 2.5 Hours, the Traveler:

For [tex]\( t \)[/tex] in the interval [tex]\( 2.5 \leq t \leq 3.5 \)[/tex], the function [tex]\( D(t) \)[/tex] is a constant value:

[tex]\[ D(t) = 875 \][/tex]
- When [tex]\( t = 2.5 \)[/tex]:

[tex]\[ D(2.5) = 875 \][/tex]

So, at 2.5 hours, the traveler is [tex]\( 875 \)[/tex] miles from home.

### 4. At 3 Hours, the Distance:

For [tex]\( t \)[/tex] in the interval [tex]\( 2.5 \leq t \leq 3.5 \)[/tex], the function [tex]\( D(t) \)[/tex] is again a constant value:

[tex]\[ D(t) = 875 \][/tex]
- When [tex]\( t = 3 \)[/tex]:

[tex]\[ D(3) = 875 \][/tex]

So, at 3 hours, the traveler is [tex]\( 875 \)[/tex] miles from home.

Combining these results, we have:

- The starting distance is [tex]\( 125 \)[/tex] miles.
- At 2 hours, the traveler is [tex]\( 725 \)[/tex] miles from home.
- At 2.5 hours, the traveler is [tex]\( 875 \)[/tex] miles from home.
- At 3 hours, the distance is [tex]\( 875 \)[/tex] miles.

Given these findings, the three noteworthy options are:
1. The starting distance, at [tex]\( t = 0 \)[/tex], is [tex]\( 125 \)[/tex] miles.
2. At 2 hours, the traveler is [tex]\( 725 \)[/tex] miles from home.
3. At 2.5 hours, the traveler is [tex]\( 875 \)[/tex] miles from home.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.