Solve Geometry Q w/ L and c: Find R in Terms of L and c

[madgorillaz15]

Homework Statement

In the attached drawing, find R in terms of L and c. Also, at the bottom of the picture I wrote something wrong. I said c, which equals 0.5, is the arc-length of each semi-circle, but I really meant to say each quarter circle. My bad. I'm not given a number for L so that can just remain a variable.

Can anyone see the solution? I'd appreciate any help.

Homework Equations

The Attempt at a Solution

I think I need to take advantage of c somehow, but I wasn't able to figure it out. Obviously, R=(a^2 + L^2)^(1/2), so I just need to figure out a in terms of L and c (or something else)?

 

[Svein]

From your figure a=R⋅sin(c) and L=R⋅cos(c). Then...

 

[titasB]

First,

a / R = sin (c) => a = R sin (c) --- (i)

Next,

R^2 = a^2 + L^2 => a = (R^2-L^2)^1/2 ---(ii)

Equating (i) and (ii) gives: R sin (c) = (R^2-L^2)^1/2 => R = L / cos(c)

 

[madgorillaz15]

From your figure a=R⋅sin(c) and L=R⋅cos(c). Then...

Can I ask, are you treating c like an angle? Also, after speaking to a friend, I was mistaken in assuming that those arcs belonged to quarter circles. In fact, they belong to an ellipse (so the entire big arc you see is actually half an ellipse). How does that change your answer?
Thanks though.

 

[madgorillaz15]

First,

a / R = sin (c) => a = R sin (c) --- (i)

Next,

R^2 = a^2 + L^2 => a = (R^2-L^2)^1/2 ---(ii)

Equating (i) and (ii) gives: R sin (c) = (R^2-L^2)^1/2 => R = L / cos(c)

Hi titasB,
I apologize, but as in my other reply, does this answer change if the big arc is actually half an ellipse, and not have a circle as I first assumed?

 

[Svein]

Can I ask, are you treating c like an angle?

Yes. If not, we need to find some other way. If it is part of a circle (as you mentioned in your original post), then the angle is c/R. Then you get a slightly more complicated expression. From the note on your figure, c=0.5, so the angle is 0.5/R. Inserting this, you get L=R⋅cos(0.5/R) - which is fine for finding L, but complicated for finding R. On the other hand, tg(0.5/R) = a/L - which gives 0.5/R = arctg(a/L).

 

[madgorillaz15]

Yes. If not, we need to find some other way. If it is part of a circle (as you mentioned in your original post), then the angle is c/R. Then you get a slightly more complicated expression. From the note on your figure, c=0.5, so the angle is 0.5/R. Inserting this, you get L=R⋅cos(0.5/R) - which is fine for finding L, but complicated for finding R. On the other hand, tg(0.5/R) = a/L - which gives 0.5/R = arctg(a/L).

Yeah, I just checked and we aren't allowed to treat it like an angle. Furthermore, I was mistaken in saying that c was a part of a circle-it's actually an ellipse. How does that change the answer?

 

[Svein]

I was mistaken in saying that c was a part of a circle-it's actually an ellipse. How does that change the answer?

Well, you will have to state your problem again, this time more precisely. What are the known parts and what are you supposed to determine? What do you know about the ellipse? What is the significance of R (an ellipse has two radii)?