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?

- anonymous

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- 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

|dw:1354392763687:dw|

- anonymous

I didn't get it =(

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## More answers

- anonymous

Yeah, that looks right. Now you just need to make it into an equation.

- anonymous

Well, if \(d_{total} = (vt)_{total}\), and \(d_{total} = d_{up} + d_{down}\), then \(d_{total} = (vt)_{up} + (vt)_{down}\)

- anonymous

So, the answer is 36 ?

- anonymous

Said another way:
\[d_{up} = (vt)_{up}\]
\[d_{down} = (vt)_{down}\]
\[d_{up} + d_{down} = 2d = (vt)_{up} + (vt)_{down}\]

- anonymous

I haven't calculated it, so I'm not sure. One sec.

- anonymous

2.6 X +3.9(12-X)= the answer ?

- anonymous

No, the answer is not 36.

- 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

Let's call time \(t\) instead of \(x\).

- anonymous

So, since \(d_{up} = d_{down}\), \((vt)_{up} = (vt)_{down}\)

- anonymous

what is the (vt) ?

- anonymous

\(vt\) is velocity times time.

- anonymous

\(d = vt\)

- anonymous

We're just plugging in \(vt\) for \(d\).

- 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

Also, remember that \((vt)_{up}\) is the same as saying \(v_{up} \times t_{up}\). Same goes for down.

- anonymous

oh okay ,so you're saying that the Distance is the Rate*Time
which is \[d=vt\] ,right?

- anonymous

Yes, exactly.

- anonymous

We can equate rate*time up to rate*time down, because they are both the same trail (i.e., same distance).

- anonymous

That's what makes it possible to solve the equation.

- anonymous

\[2.6x+3.9(12-x) \]

- anonymous

?

- anonymous

You're adding them together. You need to make them equal to each other.

- anonymous

Okay

- anonymous

I think i got it

- anonymous

Because \(2.6x = d = 3.9(12-x)\)

- anonymous

Solve for \(x\) (time up the hill), then plug it into your original formula to solve for \(d\).

- anonymous

\[2.6x=3.9(12-x) \]
\[2.6x=46.8-3.9x \]
\[46.8=6.5x \]
\[x=7.2\]

- anonymous

Good! So what does \(x\) represent?

- anonymous

time up the hill ?

- anonymous

Yes, exactly!

- anonymous

So now you can plug it into your distance formula (remember, \(d = vt\)) and solve for \(d\).

- 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

\[d=vt\]\[3.9*4.8=18.72\]
\[d=18.72\]

- anonymous

Excellent. And, conversely, \(2.6 \times 7.2 = 18.72 \text{ km}\), so you know your answer is right.

- anonymous

Thank you so much !

- anonymous

No problem. :)

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