Let's analyze the problem in detail:
Amy hikes a total of 6 miles from a ranger station to a waterfall and then returns the same distance back to the ranger station, making a round trip of 12 miles total.
The function [tex]\( T(x) \)[/tex] is given as:
[tex]\[ T(x) = \frac{6}{x} + \frac{6}{x+1} \][/tex]
where [tex]\( x \)[/tex] represents Amy's hiking speed in miles per hour.
To understand the term [tex]\( \frac{6}{x} \)[/tex]:
1. Analyze the first term [tex]\( \frac{6}{x} \)[/tex]:
- The term [tex]\( \frac{6}{x} \)[/tex] represents the time it takes to cover the 6-mile distance at a speed of [tex]\( x \)[/tex] miles per hour.
- Time is calculated using the formula [tex]\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)[/tex].
- Since [tex]\( x \)[/tex] is Amy’s hiking speed on the way to the waterfall, [tex]\( \frac{6}{x} \)[/tex] measures the time it takes her to hike to the waterfall.
2. Analyze the second term [tex]\( \frac{6}{x+1} \)[/tex]:
- The term [tex]\( \frac{6}{x+1} \)[/tex] represents the time it takes Amy to return the 6 miles to the ranger station.
- On the return trip, Amy hikes 1 mile per hour faster, thus her speed is [tex]\( x + 1 \)[/tex] miles per hour.
- The time taken for the return trip is given by [tex]\( \frac{6}{x+1} \)[/tex].
Given this understanding, we can now evaluate the options:
- Option A: It is the time it takes Amy to hike back to the ranger station.
- This is incorrect because [tex]\( \frac{6}{x} \)[/tex] is the time taken to hike to the waterfall, not back to the ranger station.
- Option B: It is the time it takes Amy to hike to the waterfall.
- This is correct because it aligns with our understanding that [tex]\( \frac{6}{x} \)[/tex] measures the time to hike the 6 miles to the waterfall at speed [tex]\( x \)[/tex].
- Option C: It is the distance that Amy hikes to the waterfall.
- This is incorrect because [tex]\( \frac{6}{x} \)[/tex] measures time, not distance.
- Option D: It is the speed at which Amy hikes to the waterfall.
- This is incorrect because [tex]\( \frac{6}{x} \)[/tex] measures time, not speed.
Therefore, the correct phrase that best describes the term [tex]\( \frac{6}{x} \)[/tex] is:
B. It is the time it takes Amy to hike to the waterfall.
