Question

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?

Answer 1

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.

Answer 2

I didn't get it =(

Answer 3

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

Answer 4

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

Answer 5

So, the answer is 36 ?

Answer 6

Said another way:

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

Answer 7

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

Answer 8

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

Answer 9

No, the answer is not 36.

Answer 10

>.<

Answer 11

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 \(d_{up}\) = \(d_{down}\).

Answer 12

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

Answer 13

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

Answer 14

what is the (vt) ?

Answer 15

\(vt\) is velocity times time.

Answer 16

\(d = vt\)

Answer 17

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

Answer 18

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.

Answer 19

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

Answer 20

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

Answer 21

Yes, exactly.

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

?

Answer 26

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

Answer 27

Okay

Answer 28

I think i got it

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

time up the hill ?

Answer 34

Yes, exactly!

Answer 35

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

Answer 36

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.

Answer 37

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

Answer 38

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

Answer 39

Thank you so much !

Answer 40

No problem. :)