Math help, need a formula for some data on an arc

[icehorse]

I'm constructing an arc that's 72 inches long, with a height of 4 inches. An online arc calculator told me that such an arc is from a circle with a radius of 164 inches. Cool.

What I'm trying to figure out is the height of the arc as you move from left to right across the 72". For example, what would the height of the arc be if I was 6" from the end, or 12" from the end, and so on.

Thanks!

p.s. I'm using this problem to test ChatGPT - so far it's very polite, patient, and persistent but not very smart. it's making error after error and contradicting itself.

I don't know the correct answer, but I can definitely tell when chatGPT's answers are wrong

 

[icehorse]

Interesting experience. working together with the AI, I figured out how to get the values I wanted! ha!

the AI made a suggestion that made the problem easier to visualize. interestingly, while its approach was correct, it did the math wrong.

 

[Debater Slayer]

I'm constructing an arc that's 72 inches long, with a height of 4 inches. An online arc calculator told me that such an arc is from a circle with a radius of 164 inches. Cool.

What I'm trying to figure out is the height of the arc as you move from left to right across the 72". For example, what would the height of the arc be if I was 6" from the end, or 12" from the end, and so on.

Thanks!

p.s. I'm using this problem to test ChatGPT - so far it's very polite, patient, and persistent but not very smart. it's making error after error and contradicting itself.

I don't know the correct answer, but I can definitely tell when chatGPT's answers are wrong

I need to be on a PC to type this out. Give me a minute.

 

[icehorse]

Hey @Debater Slayer. thanks! i think i know how to solve the problem now. so if you want to do it for fun, i'd be interested in your approach, but don't do it if it isn't for fun

 

[icehorse]

Spoiler

i can use pythagorean to construct triangles with a hypoteneuse o 164 inches and short sides of 36 then 30 then 24 then 18 then 12 then 6 to find the height

 

[Revoltingest]

Spoiler

i can use pythagorean to construct triangles with a hypoteneuse o 164 inches and short sides of 36 then 30 then 24 then 18 then 12 then 6 to find the height

It sounds similar to a problem I had when designing
machine moving skates with a spherical bearing to
accommodate transitioning from a floor to a ramp.
Finding similar triangles is useful.

 

[Debater Slayer]

I'm constructing an arc that's 72 inches long, with a height of 4 inches. An online arc calculator told me that such an arc is from a circle with a radius of 164 inches. Cool.

What I'm trying to figure out is the height of the arc as you move from left to right across the 72". For example, what would the height of the arc be if I was 6" from the end, or 12" from the end, and so on.

Thanks!

p.s. I'm using this problem to test ChatGPT - so far it's very polite, patient, and persistent but not very smart. it's making error after error and contradicting itself.

I don't know the correct answer, but I can definitely tell when chatGPT's answers are wrong

The 72 inches is the length of the arc base, correct? When I plug these values into the formula for the sagitta (the height of the arc), 72 satisfies the equation as the length of the base (i.e., the chord), not the circumference of the circular arc.

 

[icehorse]

The 72 inches is the length of the arc base, correct? When I plug these values into the formula for the sagitta (the height of the arc), 72 satisfies the equation as the length of the base (i.e., the chord), not the circumference of the circular arc.

yes, the chord length is 72 and the arc height is 4
that should get you to a radius of about 164

 

[Debater Slayer]

What I'm trying to figure out is the height of the arc as you move from left to right across the 72". For example, what would the height of the arc be if I was 6" from the end, or 12" from the end, and so on.

Since you want to calculate the height from the end, I would subtract the 6" or 12" from 72" and then plug the remaining chord length into the formula for the sagitta (height).

Here, you will have a new chord length = 72" - 6" = 66". Now calculate the corresponding height using this formula:

Edit: Posting the link because the image isn't getting embedded:

https://wikimedia.org/api/rest_v1/media/math/render/svg/04632a621c25441091fea9bbcb01301bcd64073f

My idea is to consider each point you stop at as if it were the end of the full length of a new arc, without changing the radius, and then calculate the height accordingly, as if you had cut out the remaining part of the arc. I'm not sure how accurate this approach is, though. Let me know if it works out.

 

[icehorse]

Since you want to calculate the height from the end, I would subtract the 6" or 12" from 72" and then plug the remaining chord length into the formula for the sagitta (height).

Here, you will have a new chord length = 72" - 6" = 66". Now calculate the corresponding height using this formula:

Edit: Posting the link because the image isn't getting embedded:

https://wikimedia.org/api/rest_v1/media/math/render/svg/04632a621c25441091fea9bbcb01301bcd64073f

My idea is to consider each point you stop at as if it were the end of the full length of a new arc, without changing the radius, and then calculate the height accordingly, as if you had cut out the remaining part of the arc. I'm not sure how accurate this approach is, though. Let me know if it works out.

We really only need to find the heights for half of the arc since it's symmetrical, correct? We know that at the end of the chord, the height between the arc and the chord is 0 and we know that at the middle of the chord (36 inches). the height between the chord is 4 inches.

So all we need are the heights at 6, 12, 18, 24, and 30 inches.

So we construct a right triangle for each of those 5 heights above. The hypotenuse will always be 164, and the short side - working from the end of the chord to the center - will be 30, 24, 18, 12, 6

From there we can calculate the long side, and subtract 160 from it to get the height above the chord.

 

[Debater Slayer]

We really only need to find the heights for half of the arc since it's symmetrical, correct? We know that at the end of the chord, the height between the arc and the chord is 0 and we know that at the middle of the chord (36 inches). the height between the chord is 4 inches.

So all we need are the heights at 6, 12, 18, 24, and 30 inches.

So we construct a right triangle for each of those 5 heights above. The hypotenuse will always be 164, and the short side - working from the end of the chord to the center - will be 30, 24, 18, 12, 6

From there we can calculate the long side, and subtract 160 from it to get the height above the chord.

Something is not processing for me: how is the radius supposed to be so much longer than the chord? I'm trying to visualize it in my head, but it makes much more sense for the chord to be longer, as seen here with the chord AB:

Unless we're only talking about one half of the arc, I'm not sure how the radius is 164" but the arc is only 72".

How would you clarify the solution using the above drawing? I'm trying to get a full grasp of it, so I'd appreciate that.

 

[icehorse]

This diagram you provided is great, thanks. But imagine the chord I'm talking about slices off a much smaller slice of the circle. So in your diagram, imagine that the "h" doesn't move, but we move the chord to be above the "h". so we've made a very small slice out of the top of the circle.

 

[Debater Slayer]

This diagram you provided is great, thanks. But imagine the chord I'm talking about slices off a much smaller slice of the circle. So in your diagram, imagine that the "h" doesn't move, but we move the chord to be above the "h". so we've made a very small slice out of the top of the circle.

Thanks! That's a very helpful explanation. I get it now.

 

[exchemist]

Something is not processing for me: how is the radius supposed to be so much longer than the chord? I'm trying to visualize it in my head, but it makes much more sense for the chord to be longer, as seen here with the chord AB:

Unless we're only talking about one half of the arc, I'm not sure how the radius is 164" but the arc is only 72".

How would you clarify the solution using the above drawing? I'm trying to get a full grasp of it, so I'd appreciate that.

Presumably R-h = R Cos φ . So h = R (1-Cos φ ). Or have I made a mistake somewhere?

(Blasted software refuses to display theta symbol, which is why I've resorted to phi instead.)

And then if we call the length along the circumference from the apex c, φ in radians will be c/R, won't it?

So finally we have h =R(1- Cos c/R), unless I've screwed up somewhere...........which is far from unlikely .

 

[icehorse]

@exchemist - if i understand you correctly, you're solving for "h"? So if R is 164 and the AB chord is 72, then what do you get for h?

An online calculator said that h=4, but it would be good to confirm that.

Assuming h is 4, then I wanted to know the height of the upper half of the circle every 6 inches as we move from B to M. We know it's 160 at B and 164 at C, I want to know it's height in - steps of 6 - between B and C.

To calculate the heights, I used right triangles with a hypotenuse of 164 and a short side varying between 6 and 30 inches, since 0 and 36 are already known.

does that make sense / sound correct?

 

[exchemist]

@exchemist - if i understand you correctly, you're solving for "h"? So if R is 164 and the AB chord is 72, then what do you get for h?

An online calculator said that h=4, but it would be good to confirm that.

Assuming h is 4, then I wanted to know the height of the upper half of the circle every 6 inches as we move from B to M. We know it's 160 at B and 164 at C, I want to know it's height in - steps of 6 - between B and C.

To calculate the heights, I used right triangles with a hypotenuse of 164 and a short side varying between 6 and 30 inches, since 0 and 36 are already known.

does that make sense / sound correct?

Yes. I get 4 in. But if it's the chord you're working with we don't need the trig.

The chord is 72in so one half is 36in. The right angled triangle has one side of 36in and a hypotenuse 164 in.

So by Pythagoras' theorem the unknown side, "R-h" will be given by 36 ² + (R-h) ² = 164 ²,

so R-h = √(164²-36²) = 160.

But to get the heights of this arc at the intervals you describe we will need trig again I think, just not the way I did it. Hmm.

 

[icehorse]

good old Pythagoras !

thanks guys! I got the solution AND I learned a bit about the AI's strengths and weaknesses

 

[exchemist]

Let's see. B-M/164 = Sin φ.

good old Pythagoras !

thanks guys! I got the solution AND I learned a bit about the AI's strengths and weaknesses

I don't think we're there yet,now that I understand what you are trying to do. The arc is fixed length. I'm not sure what construction we can make that tells you the heights at intervala along BM. It's got to be related to Cos φ I think because it will be a maximum when φ is zero, but I haven't worked out how to do it.

 

[icehorse]

Let's see. B-M/164 = Sin φ.

I don't think we're there yet,now that I understand what you are trying to do. The arc is fixed length. I'm not sure what construction we can make that tells you the heights at intervala along BM. It's got to be related to Cos φ I think because it will be a maximum when φ is zero, but I haven't worked out how to do it.

hmmm..

imagine a set of right triangles where h is always the same (164)
now imagine we construct the various triangles by using various short side lengths along the O line (starting at O and extending left or right). The length of these short sides will be the same as the length along the chord line, right?

So if we take an example from near the middle of BM, we'll have a right triangle with h = 164 and the short side being, perhaps, 18. pythagagoras takes us home from there, right?

 

[exchemist]

hmmm..

imagine a set of right triangles where h is always the same (164)
now imagine we construct the various triangles by using various short side lengths along the O line (starting at O and extending left or right). The length of these short sides will be the same as the length along the chord line, right?

So if we take an example from near the middle of BM, we'll have a right triangle with h = 164 and the short side being, perhaps, 18. pythagagoras takes us home from there, right?

It's after 11pm here and I'm ceasing to think straight. I think this is one for the morning, after some breakfast including a couple of cups of tea! (Of course Polymath could do it standing on his head, but he'll have other things to do on a Friday night!)