## zello Group Title 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? one year ago one year ago

1. geoffb Group Title

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 Group Title

|dw:1354392763687:dw|

3. zello Group Title

I didn't get it =(

4. geoffb Group Title

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

5. geoffb Group Title

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

6. zello Group Title

So, the answer is 36 ?

7. geoffb Group Title

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

8. geoffb Group Title

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

9. zello Group Title

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

10. geoffb Group Title

No, the answer is not 36.

11. zello Group Title

>.<

12. geoffb Group Title

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 Group Title

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

14. geoffb Group Title

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

15. zello Group Title

what is the (vt) ?

16. geoffb Group Title

$$vt$$ is velocity times time.

17. geoffb Group Title

$$d = vt$$

18. geoffb Group Title

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

19. geoffb Group Title

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 Group Title

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

21. zello Group Title

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

22. geoffb Group Title

Yes, exactly.

23. geoffb Group Title

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

24. geoffb Group Title

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

25. zello Group Title

$2.6x+3.9(12-x)$

26. zello Group Title

?

27. geoffb Group Title

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

28. zello Group Title

Okay

29. zello Group Title

I think i got it

30. geoffb Group Title

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

31. geoffb Group Title

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

32. zello Group Title

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

33. geoffb Group Title

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

34. zello Group Title

time up the hill ?

35. geoffb Group Title

Yes, exactly!

36. geoffb Group Title

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

37. geoffb Group Title

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 Group Title

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

39. geoffb Group Title

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

40. zello Group Title

Thank you so much !

41. geoffb Group Title

No problem. :)