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?

- anonymous

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- DanJS

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

- 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

- DanJS

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

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## More answers

- DanJS

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

- DanJS

|dw:1443251516317:dw|

- DanJS

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

- DanJS

you understand the picture?

- anonymous

so would the answer be( -6,5)

- DanJS

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

- anonymous

(6-5)

- DanJS

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

- anonymous

okay

- DanJS

|dw:1443252011827:dw|

- DanJS

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

- DanJS

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

- DanJS

|dw:1443252567646:dw|

- anonymous

So would it be (5-6)?

- anonymous

(5,-6)

- anonymous

sorry this is still confusing for me _-.-

- DanJS

have you had trig yet?

- anonymous

Just starting

- DanJS

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

- anonymous

yes

- DanJS

|dw:1443253028067:dw|

- DanJS

tangent of angle A is 5/6

- 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

- DanJS

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

- DanJS

|dw:1443253419214:dw|

- 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

- 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

- 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

- 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)

- 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)

- 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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