To determine which phrase best describes the term [tex]\(\frac{6}{2}\)[/tex], we need to understand the context and details of Amy's hike, as well as the meaning of each term in the function [tex]\(T(x)\)[/tex].
Amy hikes 6 miles from the ranger station to the waterfall, and then she hikes 6 miles back. The function [tex]\(T(x) = \frac{6}{z} + \frac{6}{z+1}\)[/tex] is given to represent Amy's total time spent hiking, where [tex]\(z\)[/tex] is Amy's hiking speed in miles per hour.
Now, let's break down the terms in [tex]\(T(x)\)[/tex]:
- [tex]\(\frac{6}{z}\)[/tex] represents the time it takes Amy to hike the 6 miles to the waterfall at her speed [tex]\(z\)[/tex] miles/hour.
- [tex]\(\frac{6}{z+1}\)[/tex] represents the time it takes Amy to hike the 6 miles back to the ranger station at her speed [tex]\(z + 1\)[/tex] miles/hour (since she hikes 1 mile/hour faster on her return trip).
Given the term [tex]\(\frac{6}{2}\)[/tex]:
- [tex]\(\frac{6}{2}\)[/tex] simplifies to 3.
- To understand what this 3 represents, observe the term [tex]\(\frac{6}{z}\)[/tex] in the function [tex]\(T(x)\)[/tex]. This term represents the time it takes Amy to hike to the waterfall. If [tex]\(z = 2\)[/tex], then [tex]\(\frac{6}{z} = \frac{6}{2} = 3\)[/tex].
Thus, the value 3 in the context of [tex]\(\frac{6}{2}\)[/tex] specifically refers to the time it takes Amy to hike to the waterfall when her speed [tex]\(z = 2\)[/tex] miles/hour.
Therefore, the correct phrase that best describes the term [tex]\(\frac{6}{2}\)[/tex] is:
C. It is the time it takes Amy to hike to the waterfall.
