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Pandemonium

  • one year ago

On a movie set, an archway is modeled by the equation y = -0.5x2 + 3x, where y is the height in feet and x is the horizontal distance in feet. A laser is directed at the archway at an angle modeled by the equation -0.5x + 2.42y = 7.65 such that the beam crosses the archway at points A and B. At what height from the ground are the points A and B? A. 1.5 feet and 3.5 feet B. 1.5 feet and 4 feet C. 3.5 feet and 4 feet D. 4 feet and 4 feet

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  1. mathmate
    • one year ago
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    Have you started thinking about It?

  2. Pandemonium
    • one year ago
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    ive tried solving it but i dont understand on how to start this

  3. mathmate
    • one year ago
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    To solve ANY math word problem, you go with the first step 1. read the question, understand everything that is supplied. Read it three times if necessary to understand the question perfectly. Then write a summary of the given information. Can you do that now?

  4. Pandemonium
    • one year ago
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    so look at the equation given and then read the question and what i need to find?

  5. mathmate
    • one year ago
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    After having fully understood the question, then 1. write a summary of the given information

  6. Pandemonium
    • one year ago
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    ok

  7. Pandemonium
    • one year ago
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    So i understand the question and its asking for me to find the height of a cross beam that crosses through points A and B

  8. mathmate
    • one year ago
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    I would put the summary like this: Given: Equation of arch : y=-0.5x^2+3x Equation of beam: -0.5x + 2.42y = 7.65, or \(y=\frac{0.5x+7.65}{2.42} Need to find intersection points of arch and beam

  9. mathmate
    • one year ago
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    * \( \Large y=\frac{0.5x+7.65}{2.42}\)

  10. mathmate
    • one year ago
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    This way, you have a clear mind of what is mathematically needed, and don't have to worry about wordings. But do check that you have copied all the numbers correctly. From now on until the answer line, you are working with mathematics. The answer line is the \(last\) sentence of your solution that interprets the results of math calculations in the terms of the question.

  11. Pandemonium
    • one year ago
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    alright i have written down all the numbers correctly

  12. Pandemonium
    • one year ago
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    So from here on until we get to the answer line, will we only work with mathematics?

  13. mathmate
    • one year ago
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    Then we can proceed to the second step 2. Graph the curves/lines. Use a graphics calculator if your school allows it. If not, even better, use a sheet of paper and pencil. Can you draw that for me (use the "draw" button below). This step will help you find approximately the required answer, or check the final answer for gross mistakes.

  14. Pandemonium
    • one year ago
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    alright so what exactly am i suppose to draw?

  15. mathmate
    • one year ago
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    Draw the arch and the beam, approximately.

  16. Pandemonium
    • one year ago
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    so is it a triangle type or square?

  17. Pandemonium
    • one year ago
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    |dw:1434978937794:dw|

  18. Pandemonium
    • one year ago
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    Like that?

  19. mathmate
    • one year ago
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    An arch is a curve, and the beam is a line. Are you familiar with quadratic functions and linear functions?

  20. Pandemonium
    • one year ago
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    yea im familiar with it but i could never really understand it

  21. Pandemonium
    • one year ago
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    |dw:1434979035876:dw|

  22. mathmate
    • one year ago
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    That's the idea, but we have to put them in the context of the question.

  23. Pandemonium
    • one year ago
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    and would i do that?

  24. mathmate
    • one year ago
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    First, examine the equations. The arch is a quadratic function y=-0.5x^2+3x the leading coefficient, or the coefficient of x^2, is -0.5. The negative sign means that the function is concave down, the way you drew it. The constant term (with no variable) is zero, means the function passes through the origin. If you complete the square, meaning writing it as y=-0.5(x-3)^2+4.5 It means that the maximum of the function is at x=3. So we end up with a graph like this: |dw:1434979494256:dw|

  25. mathmate
    • one year ago
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    Is that clear so far?

  26. Pandemonium
    • one year ago
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    yea im learning this now :D

  27. mathmate
    • one year ago
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    Good! Now we'll work on the beam. \(\Large y=\frac{0.5x+7.65}{2.42}\)

  28. Pandemonium
    • one year ago
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    so we plug in something to that x and solve it?

  29. mathmate
    • one year ago
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    Kind of, but we will first write this in slope intercept form and get rid of the fractions!

  30. Pandemonium
    • one year ago
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    ok

  31. mathmate
    • one year ago
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    So that makes the equation \(approximately\) y=0.207x+3.161

  32. Pandemonium
    • one year ago
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    ok then whats next?

  33. mathmate
    • one year ago
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    Can you sketch the line on the graph?

  34. mathmate
    • one year ago
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    I'll give you something for the vertical scale: |dw:1434979999842:dw|

  35. mathmate
    • one year ago
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    yes!

  36. mathmate
    • one year ago
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    Yes, these are the points we're looking for.

  37. mathmate
    • one year ago
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    Recall that in the equation y=0.207x+3.161 3.161 is the y-intercept 0.207 is the slope.

  38. Pandemonium
    • one year ago
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    so i draw i line from the 3 up and somewhere in between the 4.5 and 0 across?

  39. mathmate
    • one year ago
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    What you can do is to find the point where x=6, and joint the point with the y-intercept. Point at y-intercept (0,3.161), and the point at x=6 has y=0.207*6+3.161=4.4 So you just need to join the points (0,3.161) and (6,4.4) to locate the beam.

  40. Pandemonium
    • one year ago
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    |dw:1434980631856:dw|

  41. Pandemonium
    • one year ago
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    like that?

  42. mathmate
    • one year ago
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    Not quite. The slope is positive 0.2, so the line cannot be decreasing. Also, the point on the right is (6,4.4), you have drawn (6,0) :( |dw:1434980741403:dw| I have drawn the dots, now you do "connect the dots". :)

  43. Pandemonium
    • one year ago
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    |dw:1434980772402:dw|

  44. Pandemonium
    • one year ago
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    i saw where i made the mistake Sorry bout that

  45. mathmate
    • one year ago
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    That looks good! Can you now estimate the values of x at the intersection points?

  46. Pandemonium
    • one year ago
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    |dw:1434980899618:dw| Are those the intersection points?

  47. mathmate
    • one year ago
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    Exactly! |dw:1434980969153:dw| Can you now read off the values of x at the intersection points, approximately, of course.

  48. Pandemonium
    • one year ago
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    ok so my estimate would be 1.4, 3.3 and then 3.9,4.1

  49. mathmate
    • one year ago
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    Good, so your estimates for would be 1.4 and 3.9, right?

  50. Pandemonium
    • one year ago
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    yea

  51. Pandemonium
    • one year ago
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    and thats close to the answer 1.5 and 4 feet

  52. mathmate
    • one year ago
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    You can always continue to solve algebraically the solution (which turn out to be close to 1.5765 and 4.0102) but the graphical solution is good enough for you to choose the multiple choice answers, right?

  53. Pandemonium
    • one year ago
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    yea

  54. Pandemonium
    • one year ago
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    so the answer should be B if im correct

  55. mathmate
    • one year ago
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    The graphical solution method is very powerful. Once you master it, you can solve non-linear problems like this in one, maybe 2 minutes!

  56. Pandemonium
    • one year ago
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    hahaha this one took a long time :D

  57. mathmate
    • one year ago
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    Can you please repeat the answer with the values instead of the letters?

  58. Pandemonium
    • one year ago
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    B. 1.5 feet and 4 feet

  59. mathmate
    • one year ago
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    Yes, that is correct, since this one is the closest to your estimate of 1.4 and 3.9, right?

  60. Pandemonium
    • one year ago
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    yea

  61. mathmate
    • one year ago
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    Hope you practise the graphics method more when you have a chance!

  62. Pandemonium
    • one year ago
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    i will thanks so much @mathmate

  63. mathmate
    • one year ago
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    You're welcome, Pande! :)

  64. Pandemonium
    • one year ago
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    you helped me alot to learn that and get me a better understanding of it :D

  65. mathmate
    • one year ago
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    Yes, understanding is the key to learning math! Well done!

  66. Pandemonium
    • one year ago
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    ok im closing this down bc i have one question to do and this one involves linear-quadratic system

  67. mathmate
    • one year ago
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    You can try it a similar way!

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