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Nicole hiked up a mountain trail and camped overnight at the top. The next day she returned down the same trail. Her average rate traveling uphill was 2.6 kilometers per hour and her average rate downhill was 3.9 kilometers per hour. If she spent a total of 12 hours hiking, how long was the trail?

Mathematics
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You know the trail is the same length going up as it is going down. Therefore, her total distance travelled is \(2d\). \[ v = \frac{d}{t}\] \[t_{\text{up}} = 12 - t_{\text{down}}\] This should get you started.
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I didn't get it =(

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Other answers:

Yeah, that looks right. Now you just need to make it into an equation.
Well, if \(d_{total} = (vt)_{total}\), and \(d_{total} = d_{up} + d_{down}\), then \(d_{total} = (vt)_{up} + (vt)_{down}\)
So, the answer is 36 ?
Said another way: \[d_{up} = (vt)_{up}\] \[d_{down} = (vt)_{down}\] \[d_{up} + d_{down} = 2d = (vt)_{up} + (vt)_{down}\]
I haven't calculated it, so I'm not sure. One sec.
2.6 X +3.9(12-X)= the answer ?
No, the answer is not 36.
>.<
Okay, let's start over. First, you know that the distance going up is the same as it is going down. Therefore, you can make \(d_{up}\) = \(d_{down}\).
Let's call time \(t\) instead of \(x\).
So, since \(d_{up} = d_{down}\), \((vt)_{up} = (vt)_{down}\)
what is the (vt) ?
\(vt\) is velocity times time.
\(d = vt\)
We're just plugging in \(vt\) for \(d\).
Let's use the formula \((vt)_{up} = (vt)_{down}\). Can you plug in your values? Let's say that \(t\) represents time going up. Therefore, \(12 - t\) would be time going down.
Also, remember that \((vt)_{up}\) is the same as saying \(v_{up} \times t_{up}\). Same goes for down.
oh okay ,so you're saying that the Distance is the Rate*Time which is \[d=vt\] ,right?
Yes, exactly.
We can equate rate*time up to rate*time down, because they are both the same trail (i.e., same distance).
That's what makes it possible to solve the equation.
\[2.6x+3.9(12-x) \]
?
You're adding them together. You need to make them equal to each other.
Okay
I think i got it
Because \(2.6x = d = 3.9(12-x)\)
Solve for \(x\) (time up the hill), then plug it into your original formula to solve for \(d\).
\[2.6x=3.9(12-x) \] \[2.6x=46.8-3.9x \] \[46.8=6.5x \] \[x=7.2\]
Good! So what does \(x\) represent?
time up the hill ?
Yes, exactly!
So now you can plug it into your distance formula (remember, \(d = vt\)) and solve for \(d\).
You also know that time down the hill is \(12 - x\), which you can calculate now to 4.8. You can plug *that* into your downhill formula, and you should get the same answer for \(d\). That will help prove your answer is right.
\[d=vt\]\[3.9*4.8=18.72\] \[d=18.72\]
Excellent. And, conversely, \(2.6 \times 7.2 = 18.72 \text{ km}\), so you know your answer is right.
Thank you so much !
No problem. :)

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