## 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?

1. DanJS

do you know how to determine the equation of a circle given the center point and a point on the circle?

2. DanJS

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

centered at the origin , and the radius is the distance from (0,0) to (5,6) right?

4. DanJS

actually, here is a simple way to see the solution ....

5. DanJS

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6. DanJS

since the angle of rotation is a nice one, it makes it pretty simple

7. DanJS

you understand the picture?

8. anonymous

so would the answer be( -6,5)

9. DanJS

almost, since it travels 30/40 of the way around, or 270 degrees, you end here...

10. anonymous

(6-5)

11. DanJS

sorry, you start moving at angle A, not at angle zero, let me redo that

12. anonymous

okay

13. DanJS

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14. DanJS

the overall angle after moving is 270 plus the initial angle A

15. DanJS

the whole circle totals 360, so you can show that the remaining distance to the + x axis is 90 - A

16. DanJS

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17. anonymous

So would it be (5-6)?

18. anonymous

(5,-6)

19. anonymous

sorry this is still confusing for me _-.-

20. DanJS

21. anonymous

Just starting

22. DanJS

the tangent of an angle in a right triangle is the side opposite divided by the adjacent side...remember that?

23. anonymous

yes

24. DanJS

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25. DanJS

tangent of angle A is 5/6

26. DanJS

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

th eside lengths of the triangle are the components of the coordinate (5,6)

28. DanJS

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29. DanJS

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

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

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

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

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

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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