Select the correct answer.

Arry hikes 6 miles from a ranger station to a waterfall and back again. She hikes 1 mile per hour faster on the way back. Function [tex]T[/tex] represents Amy's total time spent hiking, and [tex]x[/tex] represents Arry's hiking speed in miles per hour.

[tex]\[ T(x)=\frac{6}{x}+\frac{6}{x+1} \][/tex]

Which phrase best describes the term [tex]\frac{6}{x}[/tex]?

A. It is the speed at which Arry hikes to the waterfall.

B. It is the distance that Arry hikes to the waterfall.

C. It is the time it takes Arry to hike back to the ranger station.

D. It is the time it takes Arry to hike to the waterfall.



Answer :

To determine the correct description for the term [tex]\(\frac{9}{x}\)[/tex] in the context of the problem involving Amy's hike, we should examine the given function for total time, [tex]\( T(x) = \frac{6}{x} + \frac{6}{x+1} \)[/tex], as well as understand what each of these terms represents:

1. [tex]\(\frac{6}{x}\)[/tex] represents the time it takes Amy to hike to the waterfall, where [tex]\(x\)[/tex] is Amy’s hiking speed in miles per hour.
2. [tex]\(\frac{6}{x+1}\)[/tex] represents the time it takes Amy to hike back to the ranger station, given that her speed on the way back is [tex]\(x + 1\)[/tex] miles per hour.

Now, let's analyze the given options in relation to [tex]\(\frac{9}{x}\)[/tex]:

A. Speed at which Amy hikes to the waterfall:
- The speed at which Amy hikes to the waterfall is [tex]\(x\)[/tex] miles per hour, not [tex]\(\frac{9}{x}\)[/tex].

B. Distance that Amy hikes to the waterfall:
- The distance Amy hikes to the waterfall is 6 miles, not [tex]\(\frac{9}{x}\)[/tex].

C. Time it takes Amy to hike back to the ranger station:
- The time it takes Amy to hike back is [tex]\(\frac{6}{x+1}\)[/tex], not [tex]\(\frac{9}{x}\)[/tex].

D. Time it takes Amy to hike to the waterfall:
- The time it takes Amy to hike to the waterfall is [tex]\(\frac{6}{x}\)[/tex], not [tex]\(\frac{9}{x}\)[/tex].

Given all the options, none of them describe the term [tex]\(\frac{9}{x}\)[/tex] accurately. Thus, based on the problem's context and the answer calculation, the correct interpretation is that none of the provided options directly describe [tex]\(\frac{9}{x}\)[/tex]. Therefore, the term [tex]\(\frac{9}{x}\)[/tex] does not correspond to any of the given choices correctly.

The correct conclusion here is that none of the given options (A, B, C, D) are a correct description of the term [tex]\(\frac{9}{x}\)[/tex].