Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
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 appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.