Math/Trigonometry Question

[LaggyLou]

Hmmmm......maybe I'm not explaining it right. Here is the problem, and what I understand Evan's solution to be:
[image]http://img175.echo.cx/my.php?image=arcproblem5pq.png[/image]

Hmmm. I Can't figure out why the image tag isn't working. Try it as a
link.

[Evan]

If you read my post where I said that divding by 2B was an error, forget that. It is correct. I just suck at checking simple math (I also suck at mixing up radius and diameter [img]/images/graemlins/shocked.gif[/img]). The correct formula is the one in my first post.

[purnell]

"Arc Length = Arc Radius * Angle in Radians "

I re-learned something today. [img]/images/graemlins/cool.gif[/img]

[LaggyLou]

Ok, I'm doing something wrong. Let's call X half the chord length and y the arc height. As I understand it:

R=(x^2+y^2)/y -- is that right?

Then:

Angle = 2*Arcsin(x/arc Radius) {Chord length = 2x, so the 2's in your formula cancel out, right?}

Arc Length = R * Angle


.... Ok, so here are some real numbers:

x=.419, y=.251

Radius = .9504
Angle = .9131, yielding
Arc Length = .8678

But I don't see how that could be right, because the length of the hypotnuse of a rt triangle with X and y as the legs is .488428. The arc length has to be more than double that hypontuse, doesn't it?

[Evan]

Yea, I edited my "correction" post. Sorry for the confusing. Your first interpretation, the on in the image you posted, is correct. I blame whoever questioned why you dived by 2B. [img]/images/graemlins/tongue.gif[/img]

As for an answer to that question, I don't know. I'm sure it's not incredibly hard to drive, but I am half studying for a final, so I can't do it right now.

[RacersEdge]

I think that drawing is right.

x = 1/2 cord lenght = known
c = cord height = known

In drawing, r (radius) = y + c

x^2 + y^2 = r^2

x^2 + (r-c)^2 = r^2

Now you know r since x,c are known.

sin(angle) = x/r

arc length = (angle/360)*2*pi*r

[tallstack]

I am responding to your calcs with reference to Tbag's diagram. Evan's equations are correct and you do need to divide by 2b (or 2y using your variables)

From TBag's diagram, we do not know the distance from the center of the circle to the chord, y. We know the distance from the chord to the outside of the circle. So in his diagram, y should be replaced with R-y.

If you use TBag's diagram then the correct equation is
R^2 = (r-y)^2 + x^2
or R^2 = (R^2 - 2Ry + y^2) + x^2
or 2Ry = y^2 + x^2
or R = (y^2 + x^2) / 2y

From your numbers,
x=.419, y=.251

Radius = .4752
Angle = 2.1589 rad
Arc Length = 1.0259

You are right that the arc length has to be more than twice the hypotenuse.

I hope that this helps.

Dave

[mostsmooth]

[ QUOTE ]
This is wrong.

[/ QUOTE ]
its not wrong if the chord goes through the center [img]/images/graemlins/cool.gif[/img]