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A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

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A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

942 inches? actually half that 471 inches

Edited by slight
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A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

The area that the tape makes in a cross section is

pi (2)2 - pi (1)2 = 3 pi

The length of the tape is then 3pi*50 = 150 pi = 471 in

Edited by bushindo
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737.7559108 inches

[pi * (1)^2] + [pi * (1.01)^2] + .... + [pi * (2)^2]

A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?
Edited by psycho
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737.7559108 inches

[pi * (1)^2] + [pi * (1.01)^2] + .... + [pi * (2)^2]

If you're integrating the length of the tape as the sum of successive concentric circles, shouldn't you use the fomula for the circumfence 2*pi*r as opposed to the area of the circle pi * r2?

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Aww that was too easy :rolleyes:.

slight & bushindo got it.

only easy if you happen to know the formula...and I messed up the calculations the first time and got the answer as double the length. considering I had to edit, this one goes to bushindo.

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Aww that was too easy :rolleyes:.

slight & bushindo got it.

But isn't the length of the tape influenced by the slow change in diameter? On the first wrap the diameter is 2, with a radius of 1 and the circumferance 3.141592654, the length of the tape at this point. The second wrap the diameter is now 2.02, or 1.01 radius and the tape would be 3.204738666 inches. Add the two together and the tape is 6.34633132 inches long at this point. Keep going until you get to the diameter of 4 or radius of 2 and you get 740.8975035 inches.

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If you're integrating the length of the tape as the sum of successive concentric circles, shouldn't you use the fomula for the circumfence 2*pi*r as opposed to the area of the circle pi * r2?

951.902574 inches.

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951.902574 inches.

OK so as a series:

The tape wraps round 50 times.

So...Length = 2pi * (1) + 2pi * (1.02) + 2pi * (1.04) + ... + 2pi * (2)

= 2pi * ( 50 + (0.02 + 0.04 + 0.06 + ... + 1)

= 2pi * (50 + 25)

= 150pi

= 471

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i might be wrong but the tape goes all the way around that way increasing the diameter each time by .04 or the radius by .02 therefore

2pi(riemann sum(1+.02x))50>=x>=0)

which equals 480.663

i think im right but id appreciate someone checking on it

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yep just checked it you guys are off because the sum of (1+1.02+ 1.04 +...+1.98+2)=49*1+2 +1*25+.5=76.5

obviously the 49 ones and the one 2 are the whole numbers in the decimals the 25 is just because .02 +.98=1 and .04+.96=1 and so on in 25 pairs and then the .5 in the center has no pair

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aprox 471 inches

quick and dirty; the average circumference is 9.42 inches ( 6.28+12.560)/2. Tape goes around 50 times ( one inch added on each side for total of 2 inch increase in diameter). Thus approximation of 471 inches

Edited by Karchi
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wow i hate doing this but were all wrong it doesnt rap around when the radius is 2 so its actually
2(74.5)pi

Considering the inner side of the tape: 2*pi*(50+1225/50)= 2*pi*74.5 = 468.10 inches

Considering the outer side of the tape: 2*pi*(50+1275/50) = 2*pi*(75.5) = 474.38 inches

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A reel is 2 inches in diameter. A thin tape (1/50 of an inch in thickness) is wrapped round the reel until the end of the tape is reached. The diameter of the reel and tape combined is now 4 inches. How long is the tape?

The first time you reel the tape around a 2in diameter, yo'll only need the tape to be = pi * 2. However, on the second lap, you'll need to consider that the diameter has increased in 2 times the thickness of the combined item, therefore for the second lap only you'll need an extra lenght of = pi * (2+(1/50)*2) So, you keep on increasing tikness till you have a combined thickness of 4. Following this steps I believe the answwer to be 468.10 more or less... depending on how many pi digits you use.

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[spoiler='Or we could take the physicist's approach

'], and just say, "how much area consists of tape, how long a rectangle would the tape be if the thickness were 1/50?"

The area of the spool is pi.

The area of the spool with tape on it is 4*pi.

The area of the tape alone is 4*pi - pi = 3*pi

If that area were a long skinny rectangle of thickness 1/50, its length would be 3*pi / (1/50) = 150*pi ~= 471.xxx inches

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wow i hate doing this but were all wrong it doesnt rap around when the radius is 2 so its actually
2(74.5)pi

I think the reason you are off a bit here is because the true length of the tape must be measured at the center of its thickness. If you just consider the first wrap around the 2" reel the true length will be (2+.02)*pi. If you continue with this process of adding 50 total wraps you will come up with 150 * pi.

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I think the reason you are off a bit here is because the true length of the tape must be measured at the center of its thickness. If you just consider the first wrap around the 2" reel the true length will be (2+.02)*pi. If you continue with this process of adding 50 total wraps you will come up with 150 * pi.

Yes that makes sense. And gives the right answer too.

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okay, I'm convinced the right answer is approx. 471 Can some of you check my previous post and see where I went wrong? I'm having trouble figuring out where I lost

all those pi. I was trying to add up the circumferences, with the first wrap being C= 2pi and then adding 2pi (49 + 49/50) . But I only came out with 101.96 pi.

I'd appreciate any help-- its been awhile since math class.

Edited by chicory
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This problem can only be estimated due to many changing variables. ie: 1.tape stretching, 2.how tight the wrap is, 3. calculation based on law of averages. If you eliminate the variables this is what I calculate.

1. The inside diameter (outside diameter of spool) is 2", The outside diameter of finished tape roll is 4", a difference of 2"

2. The tape thickness is 1/50 of an inch, taking 100 (100/50) wraps to make 2" thick tape wrap.

3. Wrap #1 is circumference of the spool: = pi* diameter = (3.1415296 * 2) = 6.2830592" long

4. Wrap #100 is circumference of the total tape wrap: = pi* diameter = (3.1415296 * 4) = 12.5661184" long

5. The difference in tape length of wrap #1 and #100 is 6.2830592" which takes 100 wraps to makeup this difference. Calculating an average, the tape will increase approximately .062830592" with every completed wrap.

6. If you increase this amount for every wrap made. then total all 100 wraps. you would get approximatly 951.883489" or 79ft 3 7/8" in length. Best Guess!

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I think it is only 50 wraps because ratio of change in circumference to change in radius is 2pi

4pi= 2pi(1+n/50) because change in circumference is 1" and then 2=1+n/50, so n=50

I think my problem is in the summation.

Edited by chicory
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