# How to find length of this tape ?

##
outer diameter 5 cm

inner diameter 3 cm

thickness of tape 0.01 cm

I used MATLAB

d=5; % diameter

x=0; % length of the tape

while d > 3

x = x + pi * d; %circumference

d = d - 0.010;

end

x / 100 % answer in meter

I got 25.258404934862064

actually the right answer should be 50.26 and it looks like I got 25 half of 50. I can't find where the fail is.

outer diameter 5 cm

inner diameter 3 cm

thickness of tape 0.01 cm

I used MATLAB

d=5; % diameter

x=0; % length of the tape

while d > 3

x = x + pi * d; %circumference

d = d - 0.010;

end

x / 100 % answer in meter

I got 25.258404934862064

actually the right answer should be 50.26 and it looks like I got 25 half of 50. I can't find where the fail is.

##### 3 Answers

Had someone else look at this and they confirm my 12.7. They think that instead of diameter you should have written radius.

My programming works out the equivalent of radius as I only used one thickness when determining cycle count.

#### Explanation:

For some reason I could not fathom my why my calculation by hand gave an unrealistic answer. So I switched to writing a quick bit of code in VBA and found the answer to be

I ran it from the centre outwards. Subtracted length of 0 up to 3cm from length of 0 up to 5cm

length_outer = 5051.68098697234

length_total = 7869.68959724238

length_inner = 2818.00861027004

length_outer = 1269.20343205028

length_total = 1971.34939012759

length_inner = 702.145958077319

Manual calculation method

#### Explanation:

Have a look at https://math.stackexchange.com/questions/1633704/the-length-of-toilet-roll

and the diagram

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Look at the comments at the bottom of my other solution. One of them explains how the system behaves and thus indicates the solution method.

Assuming given diameters should be radius (makes the numbers work).

5 cm gives 500 cycles of rotation from radius = 0 to 5 cm

3 cm gives 300 cycles of rotation from radius = 0 to 3 cm

Known Circumference

Given: tape thickness

1st rotation cycle

2nd rotation cycle

3rd rotation cycle

Observing the pattern tells us that for

So the summation of circumferences is:

For the 5 cm radius we have:

For the 3 cm radius we have:

The length of tape between and including both 3 cm and 5 cm is:

#L=12.5663"m"#

Or if diameter was meant as radius

#L=50.2655"m"#

#### Explanation:

Here is another approach

Consider the following sketch

The area of tape (green area)

#A_("Tape")=pi(5/2"cm")^2-pi(3/2"cm")^2=4pi"cm"^2#

Imagine to role out the tape, we will have a rectangle (although a very thin one)

The area of this thin rectangle, must be the same as the area of the tape, and the small side of the rectangle must be the thickness of tape

Thus

#A_("Tape")=L*T=>L=A_("Tape")/T#

Using the values

#L=(4pi "cm"^2)/(0.01"cm")=400pi"cm"~~1256.63"cm"=12.5663"m"#

**If diameter was meant as radius**

#A_("Tape")=pi(5"cm")^2-pi(3"cm")^2=16pi#

Thus

#L=(16pi"cm"^2)/(0.01"cm")=1600pi"cm"~~5026.55"cm"=50.2655"m"#