## zello 2 years ago 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?

1. geoffb

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.

2. zello

|dw:1354392763687:dw|

3. zello

I didn't get it =(

4. geoffb

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

5. geoffb

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

6. zello

So, the answer is 36 ?

7. geoffb

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

8. geoffb

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

9. zello

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

10. geoffb

No, the answer is not 36.

11. zello

>.<

12. geoffb

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 $$d_{up}$$ = $$d_{down}$$.

13. geoffb

Let's call time $$t$$ instead of $$x$$.

14. geoffb

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

15. zello

what is the (vt) ?

16. geoffb

$$vt$$ is velocity times time.

17. geoffb

$$d = vt$$

18. geoffb

We're just plugging in $$vt$$ for $$d$$.

19. geoffb

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.

20. geoffb

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

21. zello

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

22. geoffb

Yes, exactly.

23. geoffb

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

24. geoffb

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

25. zello

$2.6x+3.9(12-x)$

26. zello

?

27. geoffb

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

28. zello

Okay

29. zello

I think i got it

30. geoffb

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

31. geoffb

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

32. zello

$2.6x=3.9(12-x)$ $2.6x=46.8-3.9x$ $46.8=6.5x$ $x=7.2$

33. geoffb

Good! So what does $$x$$ represent?

34. zello

time up the hill ?

35. geoffb

Yes, exactly!

36. geoffb

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

37. geoffb

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.

38. zello

$d=vt$$3.9*4.8=18.72$ $d=18.72$

39. geoffb

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

40. zello

Thank you so much !

41. geoffb

No problem. :)