Bet You Won't Believe This !
 

The long views afforded by the beautiful weather this morning reminded me of a calculation I have long intended to do. I have always wondered how much the surface of the lake curves due to the curvature of the earth.

I looked at a map, and the longest potential view I found is indicated by the red line on the satellite picture here. From the public dock in Center Harbor to the point near Robert's Cove is a distance of 16.6 miles +/-. If the earth were flat, it appears that you could see from point A to point B without obstruction.

The diameter of the earth is 7,926 miles.

If two forum members wanted to communicate by blinking flash lights at each other they could get on step ladders at the two locations. If they both claimed to an equivalent height above the beach until they could see above the water in the broads, how high would they both have to be?

On a perfectly flat day, they would each have to be almost 46 feet above the beach.

Believe it or not.

Umm... thats an awesome fact. I've often wondered the same thing! Now I have to come up with something new to ponder on the 8 hr trip up to the lake from Philly!

My recollection of the formula is (adusted from my original post)

√1.5 X (√Ht1+√Ht2) = d (Statute Miles)(Ht=height in feet).

Plugging in your heights (46 feet each) gives d=16.61 miles. Or, if 16.6 miles is the actual distance, solving for Ht gives 45.92ft.
_________________________________________

I updated the formula. My original formula rounded √1.5 down to 1.22, which is too far off. Where do you get √1.5? All the way from the original Pythagoras' Theorem that forms the basis of the calculation. The formula multiplies Ht by Diameter of the earth, and to do this, we need to convert Ht from feet to Miles. Since there are 5280 feet to a statute mile, and the earth's diameter is 7912.2 statute miles, we get 7912.2/5280 = 1.4985227. Call it 1.5, plug it back into the formula above, and get a more accurate number.

Interesting to note, though. If you add Lake Winnipesaukee's height above sea level (504 feet), the adjusted conversion is 1.4985417. Using heights of 46 feet each now results in d=16.605. Or, if we use d=16.6 miles, solving for Ht gives 45.97 ft. Much closer to Rattlesnake Guy's original statement -- and exactly his statement after rounding!

Can someone explain then, why kids at McDonalds can't count change back?:laugh:

I give thanks!
 

Thank God--and I do mean this sincerely--for you people who can really venture deeply into the world of mathematics!! I can make change, keep a budget, balance my checkbook and all...but the other stuff scares me to death! All the power to you!

Because they don't subscribe to this forum!!!! :laugh::laugh:

The answer below is a good reason, but I personally think it is to much IPod's and cell phones texting all the time even in the class rooms per my granddaughter.
Sorry for taking over a perfectly good thread as the subject is great and a new fact is going into the memory book of knowledge of Lake Winnipesaukee.:)

It's Common Sense + Uncommon Cents
 

Because it won't be long before there won't be a need for coins? :confused:

Don't be too impressed with mathematicians—it's all made-up stuff. :mad:

Besides, anyone can look outside and see that the world is flat. :)

"NOT" ...In fact, if the forum members were about 6 feet tall, they wouldn't need ladders. Over 16.6 miles, the earth will curve 11.039 feet. Check your math!;)

Think about it. If each person was 46' above the shore, that would equate to over 5 feet of curvature per mile. (92 ft/16.6 mi = 5.5 ft/mi) If that were the case, a person standing on the northern tip of Rattlesnake Island wouldn't be able see a 10 foot high boat along the shore of Welch Island, 3 miles away.

Just for fun
 

I like to give the the folks behind the register odd amounts of change that will result in more convenient change back for me. Say 3 quarters back. The looks I get are priceless. Then when the computer tells them $3.75 change due, they nod their head.

Long before the Ipod or cell phone, RGal was working in a diner in 1982. One of the co-workers asked her,
"how much for a piece of pie?"
"$0.75"
"so...how much for two pieces of pie?"

Perhaps not as common as today.

The slide rule was one it's death bed when I was in high school. I think America would have been well served to keep it around a bit longer as a teaching aid. You had to have a feeling for what the answer would be to keep the number of decimal points correct. I am sure many of us remember it being illegal to have a calculator in school. I agree that kids need to also know how to use a calculator very well. But they also should need to know how to do it just as well in their heads.

Special thanks to This'n That for quieting my greatest fear, that I might have had the wrong answer. I have a hard time with the 46 foot step ladder.

Rattlesnake Guy's math is quite correct. You'll need to show your exact formulas if you want to challenge him, IMO.
___________________________
Actually, the earth's curvature is an integral part of the calculations, and is represented by the earth's Diameter in the formula. The Line of Sight calculation is really simply a tangental formula. Once the tangent is marked along the curvature, the actual formula is the simple Pythagorean algorithm.

Yeah, well, you forced me to unroll my ball of twine -- all 16.605 miles of it.

I'll hold the end of the twine. You go for the ride down the lake!

OK. Tail between my legs.

I used the formula to solve the distance to the horizon over 1 mile. I then (incorrectly) applied the curvature per mile on a linear basis. The earth isn't linear.

... off to McDonalds to fill out my job app.

Ironfish,

Welcome to the FORUM.

Two threads in one........I like it!
The other day at Shaws in Laconia,my bill was $16.16........so I gave the girl a $20 bill plus a quarter and a penny so that I'd get a dime back with my change.She got totally flustered and actually called her mgr over to straighten out the register...I was just chuckling to myself......great job in the schools....let them keep using those calculators.

Yeah, kinda like ...... opposite ends of a very long (16.605 miles, to be exact) string of twine.

Uhh ... yeah ... educational so far!:emb:

Looking straight down that red line view from the second floor window of Bob Bahre's house on top of Clay Point, the weekly specials posted in the window at Heath's in Centre Harbor are readable with a very cheap telescope.

....wifey to bob.....oh Bob....looks like Heaths has a good truckload special on NY Sirloin....just 1.99/lb.....should be a worth the trip to load up the boat! :D:D:D

Site distance
 

Actually,
Iron fish and Rattlesnake guy's formulas are both incorrect as neither takes into account the refraction constant. While the earths curvature makes the horizon appear slightly higher, the light moving through the earths atmosphere makes the horizon look slightly lower. I am a land surveyor and must make both corrections to every distance I measure (although my instrument makes these calculations so there is no need for tedious formulas !!):rolleye2:

They are not incorrect. They are approximations. The complete explanation of the approximations has not been provided. They could be explained -- but I doubt this FORUM is the place to do so.

But it's true that refraction is one of the things left out. Another thing left out is the complete Pythagorean formula. That was done because of the relative difference between the Earth's Diameter (7,900+ miles) and the height (46 feet) of the observers. In addition, the Earth's diameter is not the same at every point on Earth -- so we only used an average. But each of these things have a minor computational effect on the final answer. We're interested in whether the answer is 46 feet or 102 feet -- not whether it's 46 ft or 46.3 ft.

So I think it's a stretch to come in and say "these calculations are incorrect". They're a damn good approximation -- and that's all we're looking for here on the FORUM, I believe.

Great thread
 

Great thread, it is nice to see fellow math geeks showing their stuff.
Makes me wish I had stayed in a field that would have required me to stay current and use what I learned more often, nothing a quick sit down with the physics book wouln't bring back, but you get my point.

My Calculus Professor in college would still only use the slide rule (when she needed it) to perform her calculations, her statement to us at the beginning of the class was if you intend to use a calculator bring in the manual to tell you how to use it. Very smart person, the funniest thing I remember about her is that while at Princeton she needed a part-time job so she got a job crunching numbers for NASA, it turns out when she was complete they told her she had been doing the hard math to develop the polymer that coats the tiles of the most current style of space craft. It is very impressive to see someone do advanced Calculus out on the chalk board as fast as she could write without slowing down...That board is full slide the next one along, very cool.

Its funny because I used to get hounded all the time in high school for not showing my work on Algebra calcs, and it wasn't until I phycally did the math infront of my teacher without using the calculator or writing out my math did I receive a pass on having to show the work. The unfortunately thing is not everyone can do that and with "No Child Left Behind", our public schools will not produce the full potential of the students that can, and the ones that can't are even more handicapped when it comes to math.

From the helm of my boat, I can see far enough to miss rocks, other boats, people, and most anything else! That is a neat fact...I checked my iPhone and wasn't able to find an App for it! Go Sox!!

How about if only one guy goes up a ladder? I think I remember this to be approximately 4/7 of the square of the distance in miles; giving you the number of feet above the water one person would have to be to see the other at water level. This comes out to 157 feet! That is, (16.6*16.6)*4/7 = 157 feet (rounded). Does that seem right to those of you who have real formulas and not a vague recollection like I have above?

Trigonometry
 

Well I don't recall any of the formulae ( ;) ) so I derived the answers from the trigonometry. I split the difference between the radii given so far and used RG's 16.6 miles for the arc length. I got 45.9 ft for each ladder (the original problem) and 183.7 ft for just one ladder (your question). If the teacher wants, I'll submit my derivation though it's better done with drawings and that may take a bit to do here and now.

EDIT : OK teach, done !

Math
 

:confused: My head hurts reading this thread :confused:

Never a mathematician here. In my grad class we took a right brain left brain quiz one day. I tested way off the right brain charts!!! :laugh: Obviously math was never a class that I enjoyed.

Actually, this is very simple. See my equation above:

√1.5 X (√Ht1+√Ht2) = d (Statute Miles)(Ht=height in feet).

What happens when there is only one person? Well, set Ht2 = 0 in my equation, and you have your formula for one person!

It turns out that one person would have to have a ladder of 183.7 feet, just like Mee-n-Mac said. Mee-n-Mac, however, obviously likes more complicated formulas:eek:. However, my formula is full of simplifiations and approximations -- and Mee-n-Mac's efforts just show that the simplifications are justified -- in this case.

P.S. Mee-n-Mac: Angle CAF is a right angle (it's a tangent). Likewise CM(rB). That's why the simple Pthagorean algorithm works.

P.P.S. Rattlesnake Guy: You're gonna have to get a bigger ladder.

Help! My brain...it's melting!! :D

Seriously, you math guys are putting a lot of time into this, and it certainly is very interesting. More than I could do, that's for sure.

Les,
You have discovered my original motivation for the thread. Last fall, we were passing "the estate" and I wondered how far down the lake they could actually see the water. We took some photos to try and judge the height above the water. All was going well until I drew a line on the map and realized that you run into an island with this line of sight. That is why I decided to change the premise.

Thanks for all the participation. I was thinking that on the Winnipesaukee forum, even math is lake related.

I picture some poor kid using goggle for his homework and ends up here.:laugh:

I read somewhere that Bahre is going to puchase the islands in the way of his view. His intentions are to cut down the trees and demolish the island into big stone fields spread into the lake. All to increase his views down the lake.

:confused:Well I hate to blow your theory out of the water, but if your calculations are correct than it would take longer for you to boat from Center Harbor to long island than it would from long island to Roberts cove because you would be going “uphill” from center harbor to long island and “downhill” from long island to Roberts cove. And from experience this is not true!
Also if you had a water level 16.6 miles long, and held one end at center harbor and the other at Roberts cove the water level would be perfectly equal, or level. So in conclusion the earth is flat! Not round as previously stated.

Smith Point brings up an interesting point.I always thought of water as being the most accurate level.Over a long span I guess it's not but, what is level?Relative to the curvature of the earth apparently.Level actually is not a straight line then is it?Oh boy,thanks RG!

very interesting
 

Now that I have read this thread, I wonder about vision if your using it correctly or what is the vision supposed to be 20/40 20/20? Get me on a ladder that high and my blood pressure would sky rocket.

I think if the earth was flat, Rattlesnake Island would be smaller for it would have higher water level, and the state would own more property:emb:

If water was always level (flat), then picture the Atlantic Ocean from north to south. It isn't flat. I am assuming that the curvature of the earth/water has a lot to do with Earths rotation and gravitational pull, as well as the tides and gravitational pull from the Moon.

The above statement is mostly just an uneducated guess. Science experts, have fun tearing it apart! :laugh:

Earth can be modeled as an ellipsoid (since the equatorial radius is longer than the polar axis by about 23 km). This is caused by Earth's gravity and rotational forces. In the ellipsoidal model the direction of down can be shown to always be perpendicular to the ellipsoid. Thus the ellipsoid must be a surface of constant gravity potential. Fluids don't flow along the ellipsoid due to gravity. Gravity only pulls perpendicular to the ellipsoid, which is why water doesn't "flow" along the curvatures of the earth. It's not like a ball that you pour water over -- and the water flows down the sides. At any localized point, the water always only has one force -- down -- pulled by gravity.

In the real world, there are lots of other influences -- wind and currents, for example. But these are local and temporary, and don't affect the general nature of gravitational/rotational forces.

The actual story is a lot more complicated than this, of course. Scientists describe this phenomenon in terms of Geoids and Level Surfaces, for example. In fact, they call the level surface that represents mean sea level in the open ocean the Geoid. This is a surface of constant gravity potential - a level surface.

But in a word -- it's all about gravity. And don't ask me to explain gravity -- way beyond my abilities.

I don't think I'll ever get over the fact that so many people don't believe the earth is flat.

It is flat -- with respect to your local space. If you could walk on water (which, I'm sure, some FORUM members can do), and if the ocean were perfectly still, you could walk from Boston to England without ever walking uphill or downhill. 3,000 miles of comfortable, relaxing walking. In other words -- flat, with respect to gravity. :)

Straight and level ain't what it used to be
 

Correct, level and straight aren't the same thing. Even worse straight ain't what you think straight is. Imagine a perfect laser out in space far from anything. You'd think it would project a straight beam and you'd be correct. But bring that laser to Earth and it's straight line isn't the same line you'd see in far off space. Just as the Sun curves space-time so a star behind it (the famous 1919 confirmation of general relativity) the Earth will curve space-time (ever so slightly) so the laser isn't as straight as it would be in non-curved space-time. Of course the refraction effects mentioned earlier would far outweigh this effect. Good thing that I'm an engineer and not a physicist because that allows me to say "close enough".

A Joke
 

Is this what you mean, Mee-n-mac?
______________________________________
An engineer and a physicist are staying for the night in a hotel. Fortunately for this joke, a small fire breaks out in each room.

The physicist awakes, sees the fire, makes some careful observations, and on the back of the hotel's wine list does some quick calculations. Grabbing the fire extinguisher, he puts out the fire with one, short, well placed burst, and then crawls back into his dry bed and goes back to sleep.

The engineer awakes, sees the fire, makes some careful observations, and on the back of the hotel's room service list (pizza menu) does some quick calculations. Grabbing the fire extinguisher (and adding a factor of 5 for safety), he puts out the fire by hosing down the entire room several times over, and then crawls into his soggy bed and goes back to sleep.

Downhill
 

It being downhill from "the middle" to Roberts cove might just explain why the big waves seem to always come rolling down from "up there".

But how does the water get back?:rolleye1:

I've always known a VHF radio is good for about 17 miles. This is based upon the curvature of the earth and VHF signals traveling in a straight line. I believe the figure given includes a reasonable distance above the water for a boat antenna. A tall antenna sending or receiving would allow the comminucation to go through over a longer distance.

I will say, I did have an occurance where I needed to call the coast guard via VHF while off the coast of Gloucester. Funny thing was a received a reply from Coast Guard station Woods Hole(Falmouth), their antenna on a very high tower made that reception possible. They of course passed me along to Coast Guard station Boston and we worked things out from there.

So
 

This all boils down to that I am actually going up the 'saukee half way to Center Harbor and then down the 'saukee coming into Center Harbor.

How come I'm not saving any gas on the last leg? :confused:

USS Enterprise
 

And that is why Captian Kirk can bring back whales to save the Earth any "time" he wants.

Lot's of buckets
 

Why it gets back the same way it always has (as shown below). Sheeesh you island people have no idea of the effort we shore people have to go through just so the water stays where you want it .... ;)


{the end of the bucket brigade ..}