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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?

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  1. geoffb
    • 2 years ago
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    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
    • 2 years ago
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    |dw:1354392763687:dw|

  3. zello
    • 2 years ago
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    I didn't get it =(

  4. geoffb
    • 2 years ago
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    Yeah, that looks right. Now you just need to make it into an equation.

  5. geoffb
    • 2 years ago
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    Well, if \(d_{total} = (vt)_{total}\), and \(d_{total} = d_{up} + d_{down}\), then \(d_{total} = (vt)_{up} + (vt)_{down}\)

  6. zello
    • 2 years ago
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    So, the answer is 36 ?

  7. geoffb
    • 2 years ago
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    Said another way: \[d_{up} = (vt)_{up}\] \[d_{down} = (vt)_{down}\] \[d_{up} + d_{down} = 2d = (vt)_{up} + (vt)_{down}\]

  8. geoffb
    • 2 years ago
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    I haven't calculated it, so I'm not sure. One sec.

  9. zello
    • 2 years ago
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    2.6 X +3.9(12-X)= the answer ?

  10. geoffb
    • 2 years ago
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    No, the answer is not 36.

  11. zello
    • 2 years ago
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    >.<

  12. geoffb
    • 2 years ago
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    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
    • 2 years ago
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    Let's call time \(t\) instead of \(x\).

  14. geoffb
    • 2 years ago
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    So, since \(d_{up} = d_{down}\), \((vt)_{up} = (vt)_{down}\)

  15. zello
    • 2 years ago
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    what is the (vt) ?

  16. geoffb
    • 2 years ago
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    \(vt\) is velocity times time.

  17. geoffb
    • 2 years ago
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    \(d = vt\)

  18. geoffb
    • 2 years ago
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    We're just plugging in \(vt\) for \(d\).

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

  21. zello
    • 2 years ago
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    oh okay ,so you're saying that the Distance is the Rate*Time which is \[d=vt\] ,right?

  22. geoffb
    • 2 years ago
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    Yes, exactly.

  23. geoffb
    • 2 years ago
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    We can equate rate*time up to rate*time down, because they are both the same trail (i.e., same distance).

  24. geoffb
    • 2 years ago
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    That's what makes it possible to solve the equation.

  25. zello
    • 2 years ago
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    \[2.6x+3.9(12-x) \]

  26. zello
    • 2 years ago
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    ?

  27. geoffb
    • 2 years ago
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    You're adding them together. You need to make them equal to each other.

  28. zello
    • 2 years ago
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    Okay

  29. zello
    • 2 years ago
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    I think i got it

  30. geoffb
    • 2 years ago
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    Because \(2.6x = d = 3.9(12-x)\)

  31. geoffb
    • 2 years ago
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    Solve for \(x\) (time up the hill), then plug it into your original formula to solve for \(d\).

  32. zello
    • 2 years ago
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    \[2.6x=3.9(12-x) \] \[2.6x=46.8-3.9x \] \[46.8=6.5x \] \[x=7.2\]

  33. geoffb
    • 2 years ago
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    Good! So what does \(x\) represent?

  34. zello
    • 2 years ago
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    time up the hill ?

  35. geoffb
    • 2 years ago
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    Yes, exactly!

  36. geoffb
    • 2 years ago
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    So now you can plug it into your distance formula (remember, \(d = vt\)) and solve for \(d\).

  37. geoffb
    • 2 years ago
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    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
    • 2 years ago
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    \[d=vt\]\[3.9*4.8=18.72\] \[d=18.72\]

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

  40. zello
    • 2 years ago
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    Thank you so much !

  41. geoffb
    • 2 years ago
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    No problem. :)

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