anonymous
  • anonymous
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
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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.
anonymous
  • anonymous
|dw:1354392763687:dw|
anonymous
  • anonymous
I didn't get it =(

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

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