Question

Eight turns of a wire are wrapped around a pipe with a length of 20 cm and a circumference of 6cm. What is the length of the wire?

Answer 1

my guess would be to think of the pipe as being rolled out so it looks like a rectangle of length 20 and width the circumference 6.
Now each time the wire is wrapped around it travels one revolution of the pipe or 6cm but it also goes down the length of the pipe 20/8 or 2.5 cm, going back to the rectangle that would look like pulling the wire diagonally across the rectangle to a point 2.5 cm down forming a right triangle.
Find the length of the hypotenuse of that triangle which is length of the wire for first revolution and multiply by 8.
_>c^2 = 6^2+2.5^2 = 42.25
->c = 6.5
_>length of pipe = 6.5 * 8 = 52

Answer 2

hope that made sense

Answer 3

SO IF THE RECTANGLE IS ROLLED OUT WOULD THE 6 BE THE BASE

Answer 4

yes

Answer 5

wait it depends if the pipe is standing up then its the base :)

Answer 6

What would be the equation without the the numbers only using (b) (H) or whatever, so I can try it

Answer 7

Given length of pipe H, circumference C with wire wrapped N times.
length of wire = sqrt(C^2+(H/N)^2)*N

Answer 8

How did you get 6.5 c

Answer 9

sqrt(C^2+(H/N)^2), C=6,H=20,N=8

Answer 10

I forgot to sqrt it, then 52 will be the length of the wire?

Answer 11

yes

Answer 12

thanks so much

Answer 13

welcome, i dont get your railroad problem, is there more info?

Answer 14

thats was the whole question the teacher gave me, I emailed her today hopefully she get give me a hint.

Answer 15

what number did you get in the end for the length?

Answer 16

ok sounds like you need the initial height of the rail, im prob just missing something

Answer 17

I get about 507.5cm

Answer 18

how do you get that? there are only 8 turns of the wire

Answer 19

8 complete turns stretched out over 20cm of length.

Answer 20

A turn is 2pi radians.

Answer 21

It's a spiral.

Answer 22

You have to find a parametric equation for the spiral in terms of the information you have and find the arc length of the spiral.

Answer 23

ok so what equation would use to slove the question?

Answer 24

The equation for a spiral is \[u(\theta) = (r \cos \theta , r \sin \theta , a \theta + b)\]

Answer 25

I've jumped a couple of steps because I don't have much time.

Answer 26

You know what the radius of the spiral should be, so you need to find a and b. This last coordinate is the height, z. When theta =0, the height of the spiral is 0, so b is 0. When the height of the spiral is 20, theta = 8 x 2pi = 16 pi (since you're told there are 8 complete turns in 20cm), so 20 = 16pi.a --> a = 5/(4pi).

Answer 27

\[u(\theta) = (r \cos \theta, r \sin \theta , \frac{5 \pi}{4}\theta)\]

Answer 28

The arc length is given by\[L=\int\limits_{\theta = 0}^{\theta = 16 \pi}\sqrt{r^2 \cos^ 2 \theta + r^2 \sin^2 \theta + \frac{25 \theta^2}{16\pi^2}}d\theta \]\[=\int\limits_{\theta = 0}^{\theta = 16 \pi}\sqrt{r^2 + \frac{25 \theta^2}{16\pi^2}}d \theta =\frac{1}{4\pi}\int\limits_{0}^{16 \pi}\sqrt{16\pi^2 r^2+25 \theta^2}d \theta\]

Answer 29

\[r=\frac{3}{\pi}\]so substitute that in and integrate.

Answer 30

You'll have to make a substitution in your integral in order to solve it. I don't have anymore time, sorry. If this isn't something due very soon, I can come back to it.

Answer 31

hmm what class is this for?

Answer 32

Sorry if I've confused matters :s

Answer 33

no you're correct i just doubt parametric equations is required for the class

Answer 34

rosette what class was this problem for?

Answer 35

anyway 507 is obviously way too much you would have to wind the wire around the pipe like 80 times

Answer 36

this is for quantitavtive literacy