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anonymous

  • one year ago

A windmill has its center at (0, 0). Its blades make a complete rotation every 40 seconds. If the tip of one blade is positioned at (5, 6), after 30 seconds where is the tip of the blade located?

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  1. DanJS
    • one year ago
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    do you know how to determine the equation of a circle given the center point and a point on the circle?

  2. DanJS
    • one year ago
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    the general form for a circle is (x-h)^2 + (y-k)^2 = r^2 (h,k) is the center point, and r is the radius

  3. DanJS
    • one year ago
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    centered at the origin , and the radius is the distance from (0,0) to (5,6) right?

  4. DanJS
    • one year ago
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    actually, here is a simple way to see the solution ....

  5. DanJS
    • one year ago
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    |dw:1443251516317:dw|

  6. DanJS
    • one year ago
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    since the angle of rotation is a nice one, it makes it pretty simple

  7. DanJS
    • one year ago
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    you understand the picture?

  8. anonymous
    • one year ago
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    so would the answer be( -6,5)

  9. DanJS
    • one year ago
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    almost, since it travels 30/40 of the way around, or 270 degrees, you end here...

  10. anonymous
    • one year ago
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    (6-5)

  11. DanJS
    • one year ago
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    sorry, you start moving at angle A, not at angle zero, let me redo that

  12. anonymous
    • one year ago
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    okay

  13. DanJS
    • one year ago
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    |dw:1443252011827:dw|

  14. DanJS
    • one year ago
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    the overall angle after moving is 270 plus the initial angle A

  15. DanJS
    • one year ago
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    the whole circle totals 360, so you can show that the remaining distance to the + x axis is 90 - A

  16. DanJS
    • one year ago
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    |dw:1443252567646:dw|

  17. anonymous
    • one year ago
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    So would it be (5-6)?

  18. anonymous
    • one year ago
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    (5,-6)

  19. anonymous
    • one year ago
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    sorry this is still confusing for me _-.-

  20. DanJS
    • one year ago
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    have you had trig yet?

  21. anonymous
    • one year ago
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    Just starting

  22. DanJS
    • one year ago
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    the tangent of an angle in a right triangle is the side opposite divided by the adjacent side...remember that?

  23. anonymous
    • one year ago
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    yes

  24. DanJS
    • one year ago
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    |dw:1443253028067:dw|

  25. DanJS
    • one year ago
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    tangent of angle A is 5/6

  26. DanJS
    • one year ago
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    you can see the tan(angle) = y/x from that draw a vertical line from the point to the x axis, forms a right triangle

  27. DanJS
    • one year ago
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    th eside lengths of the triangle are the components of the coordinate (5,6)

  28. DanJS
    • one year ago
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    |dw:1443253419214:dw|

  29. DanJS
    • one year ago
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    Basically you have two unknown values you need to find, the x and the y of that point. To do that you need to equations that relate x and y. This tangent is one way then i will use the circle equation

  30. DanJS
    • one year ago
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    again, the tan(angle) = opposite /adjacent = y / x Angle A tan(A) = 6/5, so A = tan^(-1)(6/5) = 50.2 so (90 - A) = 39.8

  31. DanJS
    • one year ago
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    now the triangle ending on the point (x,y) Tan(90-A) = Tan(39.8056) = 0.833333333... = y/x that is the first relationship for x and y, since that is a rational decimal we can actually just finish it with this one equation... 0.8333333333.. = y/x any rational number can be converted to a ratio a/b or in this case x / y

  32. DanJS
    • one year ago
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    Convert 0.83333.... to a fraction. let - s = 0.83333... so, 10s = 8.3333.... and 100s = 83.3333... ---------------------------- 100s - 10s = 83.3333... - 8.3333... 90s = 83 - 8 90s = 75 s = 75/90 or 5/6 0.833333333 = 5/6 = y/x y = 5 and x = 6 the point is (x , y) = (6 , 5)

  33. DanJS
    • one year ago
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    keep in mind that those are the lengths of the side of the triangle, the x value is still +6, but the y value is negative there, y=-5 point with respect to xy-axis (x,y) = (6 , -5)

  34. DanJS
    • one year ago
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    if you had to, you could use a second equation with x and y unknown, the circle centered at origin, and radius is distance from origin to the given point \[r = \sqrt{5^2+6^2} = \sqrt{61}\] x^2 + y^2 = r^2 --(circle centered at the origin) \[x^2 + y^2 = [\sqrt{61}]^2 = 61\] Two equations and two unknown values x and y, you can solve it x^2 + y^2 = r^2 and tan(Angle) = y/x solvable for x and y with a known Angle

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