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freemap

  • one year ago

3. When a boat traveled downstream from Town A to Town B, the trip took 3 h. When the same boat traveled upstream from Town B to Town A, the trip took 3.6 h. For each trip, the speed of the boat and the water current were unchanged. Let x represent the speed of the boat and let y represent the speed of the water.

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  1. freemap
    • one year ago
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    (a) Write an expression for the distance traveled downstream using 3 h for the time. Then write an expression for the distance traveled upstream using 3.6 h for the time. (b) Set the expressions in Part (a) equal to each other. Then solve the equation for y. Show your work. (c) What percent of the boat’s speed is the water current?

  2. phi
    • one year ago
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    have you learned rate * time = distance or speed * time = distance or velocity * time = distance ?

  3. freemap
    • one year ago
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    yes I have

  4. phi
    • one year ago
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    For traveled downstream first, what is the speed of the boat going with the current?

  5. freemap
    • one year ago
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    I don't remember how to do this. I'm I suppose to multiply?

  6. phi
    • one year ago
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    The only reason people study math is to give them an excuse to think (or puzzle out a problem) Say you were in a canoe but did not paddle, and drifted downstream. According to the info in this problem, how fast will you be moving ?

  7. freemap
    • one year ago
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    Umm I really don't mean to sound stupid, but 1 maybe. not very fast

  8. phi
    • one year ago
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    you will be going at a speed of "y" (they tell us let y represent the speed of the water.)

  9. phi
    • one year ago
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    though y is a letter, it means "speed of the current" (it's short-hand)

  10. phi
    • one year ago
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    If you start paddling you will go faster. You would add on the speed you can paddle they say Let x represent the speed of the boat (when paddling in still water) do you know how to show x added to y ?

  11. freemap
    • one year ago
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    hmm y+ x+y?

  12. phi
    • one year ago
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    just x+y that is how fast you go if you go downstream we might want to put parens around it , so we remember it as one thing: speed of the boat: (x+y) now we use speed * time = distance (x+y)* time = distance they don't tell us the distance, we let's call the distance "d" but we know the time. can you fill in the time with a number and write the full equation?

  13. freemap
    • one year ago
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    (x+y)*1=d

  14. phi
    • one year ago
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    looks good, except why are you using 1 for the time ? what does the question say the time is for going downstream ?

  15. freemap
    • one year ago
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    (x+y)*3.6h=d

  16. phi
    • one year ago
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    Write an expression for the distance traveled downstream using 3 h for the time.

  17. freemap
    • one year ago
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    (x+y)*3h=d

  18. phi
    • one year ago
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    ok. I think the h mean hours. Probably we should leave it off. so (x+y)*3 = d or 3(x+y)= d is the answer to the first part.

  19. phi
    • one year ago
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    now you need to the speed going upstream any ideas ?

  20. freemap
    • one year ago
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    (x+y)*3.6=d or 3.6(x+y)=d

  21. phi
    • one year ago
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    (x+y) is how fast you go downstream when you go upstream, you go slower

  22. freemap
    • one year ago
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    would it be x-y

  23. phi
    • one year ago
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    the river is taking you backwards at a speed of y , as you paddle upriver at a speed of x yes (x-y) is the speed going upstream

  24. freemap
    • one year ago
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    ok would we still multiply the time like (x-y)*3.6=d

  25. phi
    • one year ago
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    yes, exactly. now you have part a) 3(x+y)= d 3.6(x-y) = d

  26. phi
    • one year ago
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    now ***Set the expressions in Part (a) equal to each other. Then solve the equation for y. Show your work. *** we see that 3(x+y) = d and 3.6(x-y) also equals d. if both are equal to d, we can say 3(x+y)= 3.6(x-y) Does that make sense ?

  27. freemap
    • one year ago
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    It does make since d is the outcome in both equations

  28. freemap
    • one year ago
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    expressions i mean

  29. phi
    • one year ago
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    to solve, distribute the 3 on the left side. that means multiply 3 times x and times y ditto for the 3.6 on the other side

  30. freemap
    • one year ago
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    ok 3x+3y=3.6x-3.6y do we then add like terms?

  31. phi
    • one year ago
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    yes,

  32. phi
    • one year ago
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    3x+3y=3.6x-3.6y I would add 3.6y to both sides 3x + 3y + 3.6y = 3.6x -3.6y +3.6y

  33. phi
    • one year ago
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    you get 3x+6.6y = 3.6x now add -3x to both sides

  34. freemap
    • one year ago
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    6.6y=-0.833

  35. phi
    • one year ago
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    you lost the x? what is 3.6x - 3x ?

  36. freemap
    • one year ago
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    0.6x

  37. phi
    • one year ago
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    yes, so you get 3x+6.6y = 3.6x 6.6y = 0.6 x now divide both sides by 6.6 what do we get ?

  38. freemap
    • one year ago
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    11

  39. phi
    • one year ago
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    you get \[ y= \frac{0.6}{6.6} x\] that simplifies to \[ y = \frac{1}{11} x \]

  40. freemap
    • one year ago
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    Ok I get it

  41. phi
    • one year ago
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    **(c) What percent of the boat’s speed is the water current? *** I would find the ratio of y/x and change it to a percent. of course, we can't use just "y", but we know y is the same as x/11 so use x/11 instead. can you do that ?

  42. phi
    • one year ago
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    you do \[ \frac{y}{x}= y \cdot \frac{1}{x} \] but y is x/11 so \[ \frac{y}{x}= \frac{x}{11} \cdot \frac{1}{x} \]

  43. freemap
    • one year ago
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    ok, so if i multiply it turns back into 1x and 11x so I'm really sure how to solve this

  44. phi
    • one year ago
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    you should learn that if you have the same thing "up top" and "below" , they cancel when you multiply fractions, you multiply top times top and bottom times bottom \[ \frac{x \cdot 1 }{11 \cdot x}\] as you know we can change the order of the multiply (right ?) so it's the same as \[ \frac{1 \cdot x }{11 \cdot x}\] but that is the same as multiplying the two fractions \[ \frac{1 \cdot x }{11 \cdot x} = \frac{1}{11} \cdot \frac{x}{x}\]

  45. phi
    • one year ago
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    the last step, we "undid" the multiply the point is, we have x/x and anything divided by itself is 1 in other words we get \[ \frac{1}{11} \cdot 1 = \frac{1}{11} \]

  46. phi
    • one year ago
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    that is the long way. the short way is to say "x up top, x down below" cross off both \[ \frac{y}{x}= \frac{\cancel{x}}{11} \cdot \frac{1}{\cancel{x}} = \frac{1}{11}\]

  47. freemap
    • one year ago
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    Thats what I meant to say 1/11 because I did cross multiply

  48. phi
    • one year ago
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    no, you should not cross multiply. Try to follow what I posted up above. meanwhile, to answer the question, change 1/11 to a decimal, and then multiply by 100 to make it a percent.

  49. freemap
    • one year ago
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    ok, 0.11

  50. phi
    • one year ago
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    0.11 means 11/100 that is different from 1/11 to change it to a decimal, divide 11 into 1 (a calculator will do that) or type 1/11= into google

  51. freemap
    • one year ago
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    9.09 after multiplying 100

  52. phi
    • one year ago
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    and add a % sign (which is how we show we multiplied by 100) 9.09%

  53. freemap
    • one year ago
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    9.09% got it

  54. freemap
    • one year ago
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    Thank you so very much. I really appreciate your help.

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