So adding 6.28 feet of rope to a circle of any arbitrary circumference, like the size of the observable universe, will increase the diameter by 2 feet? It seems wrong
You can sorta imagine like this - if you have a square with a rope on the perimeter of it, and want to raise the rope 1 foot. Since the square has 4 sides, you would have to raise each side of the rope 1 foot, and add a total of 4 feet of rope to a square.
It's been a while since my algebra classes...why does 2π(r+1) simplify to 2πr + 2π ? That is the crux of solving the equation but doesn't make any sense to me.
I have seen this "proof" before (including Wikipedia and other sites that are copying this "proof"). Despite your drawings, I believe there is a flaw in your thinking. Sorry, but my high school math says you have to convert R to an actual measurement and not just keep it as R. If you convert R1 to feet and then calculate r2 = R1+10, you will see the difference. The concept that R is irrelevant to the solution doesn't make any sense. If R was a radius of 5 feet and you extended it by 10 feet, that would be a significant difference. If R was 10000 miles, an extension of 10 feet would be very little difference. For the original band around the Earth: R1 = 3,959 miles R1 = 20,903,520 feet (3,959 miles * 5280) C1 = 2 * PI * 20,903,520 feet = 131,340,690 ft For the new expanded band C2 = C1 + 10 ft = 131,340,690 + 10 = 131,340,700 ft R2 = 131,340,700 / 2 / PI = 20,903,522 The height above the adjusted band H = R2 - R1 = 20,903,522 feet - 20,903,520 ft = 2 ft So, the new band around the Earth, after having been extended by 10 feet, distributed around the Earth would be 2 feet above the original band. But, please, don't keep perpetuating this idea that the Radius is insignificant to the solution.
The problem here is your assumption that R / (R+10) is a constant is incorrect. Note: 2 / (2+10) = 2 / 12 = 0.167. Note: 100 / (100+10) = 100 / 110 = .909. Can you do this? It is not a conundrum, just an assumption that believing that because R / (R+10) seems to be a constant, it is NOT a constant when putting real numbers in there. Also, you have to use real measurements. The usage of 6.28 feet in your video is incorrect. How did you come up with feet if you cancelled out R? Try the entire calculation using feet (convert miles to feet first), then see what you get. Sorry to burst your bubble.
he is absolutely right. The radius has absolutely non consequence. Take a radius of 10 000 feet and make it 10 001 feet, the circonference will increade by 6.2832 wich is 2pi. take a radius of 10 000 feet and make it 10 010 feet, the difference will be 62.832 feet wich is 20 pi. very simple to calculate
So adding 6.28 feet of rope to a circle of any arbitrary circumference, like the size of the observable universe, will increase the diameter by 2 feet? It seems wrong
Really hard to fathom. I find this hard to believe as well.
It seems wrong because it is wrong. See the above proof.
You can sorta imagine like this - if you have a square with a rope on the perimeter of it, and want to raise the rope 1 foot. Since the square has 4 sides, you would have to raise each side of the rope 1 foot, and add a total of 4 feet of rope to a square.
Seems impossible! So weird! And yet I can't argue with the math.
Extra length = difference in the circumferences = 2pr*R -2pi*r = 2pi(R-r)
If R - r = 1 ft, the extra l;length is 2pi(1) = 2pi.ft = 6.28ft
Wow
Yeah
It's been a while since my algebra classes...why does 2π(r+1) simplify to 2πr + 2π ? That is the crux of solving the equation but doesn't make any sense to me.
We distribute the monomial 2pi to the binomial r + 1
@@learnmathbydoing yeah, I had to go back and look at the rules for simplifying. It’s been decades!
❤️
Ok, I don't understand how 2π(r+1) simplifies to 2πr + 2π.
Distribute 2pi to r and 1. 2pi * r = 2pi r. 2pi*1 just = 2pi.
Not drawn to scale
I have seen this "proof" before (including Wikipedia and other sites that are copying this "proof"). Despite your drawings, I believe there is a flaw in your thinking. Sorry, but my high school math says you have to convert R to an actual measurement and not just keep it as R. If you convert R1 to feet and then calculate r2 = R1+10, you will see the difference. The concept that R is irrelevant to the solution doesn't make any sense. If R was a radius of 5 feet and you extended it by 10 feet, that would be a significant difference. If R was 10000 miles, an extension of 10 feet would be very little difference.
For the original band around the Earth:
R1 = 3,959 miles
R1 = 20,903,520 feet (3,959 miles * 5280)
C1 = 2 * PI * 20,903,520 feet = 131,340,690 ft
For the new expanded band
C2 = C1 + 10 ft = 131,340,690 + 10 = 131,340,700 ft
R2 = 131,340,700 / 2 / PI = 20,903,522
The height above the adjusted band
H = R2 - R1 = 20,903,522 feet - 20,903,520 ft = 2 ft
So, the new band around the Earth, after having been extended by 10 feet, distributed around the Earth would be 2 feet above the original band. But, please, don't keep perpetuating this idea that the Radius is insignificant to the solution.
The problem here is your assumption that R / (R+10) is a constant is incorrect. Note: 2 / (2+10) = 2 / 12 = 0.167. Note: 100 / (100+10) = 100 / 110 = .909. Can you do this? It is not a conundrum, just an assumption that believing that because R / (R+10) seems to be a constant, it is NOT a constant when putting real numbers in there. Also, you have to use real measurements. The usage of 6.28 feet in your video is incorrect. How did you come up with feet if you cancelled out R? Try the entire calculation using feet (convert miles to feet first), then see what you get. Sorry to burst your bubble.
Your C2 figure is wrong mate, you have to add 10ft to 20,903,520 and then multiply by 2 Pi
he is absolutely right. The radius has absolutely non consequence. Take a radius of 10 000 feet and make it 10 001 feet, the circonference will increade by 6.2832 wich is 2pi. take a radius of 10 000 feet and make it 10 010 feet, the difference will be 62.832 feet wich is 20 pi. very simple to calculate