Answered

Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

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
To solve this question, let's evaluate the traveler's distance from home using the given piecewise function [tex]\( D(t) \)[/tex] at specific time points.

1. The starting distance, at 0 hours:

For [tex]\( t = 0 \)[/tex]:
[tex]\[ D(0) = 300 \cdot 0 + 125 = 125 \][/tex]

Therefore, the starting distance is 125 miles, not 300 miles. This means the statement "The starting distance, at 0 hours, is 300 miles" is false.

2. At 2 hours, the traveler is 725 miles from home:

For [tex]\( t = 2 \)[/tex]:
[tex]\[ D(2) = 300 \cdot 2 + 125 = 600 + 125 = 725 \][/tex]

Therefore, the statement "At 2 hours, the traveler is 725 miles from home" is true.

3. At 2.5 hours, the traveler is still moving farther from home:

Just before [tex]\( t = 2.5 \)[/tex], we need to check if the function value is increasing:
- For [tex]\( t \)[/tex] values just less than 2.5, use [tex]\( D(t) = 300t + 125 \)[/tex].

As [tex]\( t \)[/tex] approaches 2.5 from the left:
[tex]\[ D(2.5 - \epsilon) = 300(2.5 - \epsilon) + 125 \][/tex]
where [tex]\( \epsilon \)[/tex] is a very small positive number.

Since [tex]\( D(t) \)[/tex] is continuous and increasing in the interval [tex]\([0, 2.5)\)[/tex], as [tex]\( t \)[/tex] approaches 2.5, [tex]\( D(t) \)[/tex] gets closer to 875.

Therefore, the statement "At 2.5 hours, the traveler is still moving farther from home" is true.

4. At 3 hours, the distance is constant, at 875 miles:

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

Therefore, the statement "At 3 hours, the distance is constant, at 875 miles" is true.

5. The total distance from home after 6 hours is 1,062.5 miles:

For [tex]\( t = 6 \)[/tex]:
[tex]\[ D(6) = 75 \cdot 6 + 612.5 = 450 + 612.5 = 1062.5 \][/tex]

Therefore, the statement "The total distance from home after 6 hours is [tex]$1,062.5$[/tex] miles" is true.

Given these evaluations, the correct statements are:

1. At 2 hours, the traveler is 725 miles from home.
2. At 2.5 hours, the traveler is still moving farther from home.
3. At 3 hours, the distance is constant, at 875 miles.
4. The total distance from home after 6 hours is [tex]$1,062.5$[/tex] miles.

The three correct options from the list are:

- At 2 hours, the traveler is 725 miles from home.
- At 2.5 hours, the traveler is still moving farther from home.
- At 3 hours, the distance is constant, at 875 miles.

Another valid option is the total distance after 6 hours being 1,062.5 miles although the question asked for three options only.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.