Math

Hi I have a cone of 100m radius and 17m height made out of a fabric of 0.01 millimeters.
I need to calculate the volume this takes when folded.
I have done this 2 different ways and got 2 different answers. I didn't expect the answers to be identical but they are too different.
Maybe someone in the forum can spot the fallacy.

method 1 - I think that this is the intended, more usual solution.

I have assumed that the cone is open. If it is not open then you will need to add the area of the circular base before you multiply by the thickness.

The material is 0.01/1000 = 0.00001m thick
If the radius of the cone is 100m and the perpendicular height is 17m
then the slant height is sqr(100 ² + 17 ²]) = sqr( 10289) = approx 101.435 metres

Surface Area of the cone = slant height * radius * pi = 101.435 * 100 * pi = approx 31,866.653 m ²

Volume = area * thickness = 31,866.653 * 0.00001 = approx 0.32 m ³

It think it would have to be very well vaccuum packed to get anywhere near this small but using this method this is about how much space the material would take up.

You may like to look at the web page below.
http://www.wikihow.com/Find-the-Surface-Area-of-Cones

Method 2

This time I found the internal volume of the pyramid and subtracted it from the external volume
I took the internal radius to be 100m and the external radius to be 100.00001m

V = (1/3) * pi * r ² * h

Internal volume
= (1/3) * pi * 100 ² * 17
=178023.5837m ³

External volume
=(1/3) * pi * 100.00001 ² * 17
=(1/3) * pi * 10000.0020000001 * 17
=178023.6190814

Difference= 178023.6190814 - 178023.5837 = 3.53 * 10 ^-2 = 0.035 m ³

I did try letting the external radius be 100m but it made very little difference to the answer.

So the first more conventional method gives a volume of 0.32m ³ and the second method gives a volume of 0.035m ³

I expected the two answers to be almost identical. Can someone see where my logic or my calculations are at fault and explain it to me please.

Sorry I meant to say 100m diameter, but I can just cut the radius in half then
Much obliged that was a very well detailed explanation!

SunRedux:

Sorry I meant to say 100m diameter, but I can just cut the radius in half then
Much obliged that was a very well detailed explanation!

Also yes the cone is open, very much like a funnel made of fabric!

Okay,
I've had a think a and a chat to another and I can account for at least some of the error in my second method solution.

First I should have had the external height as 17.00001m
and
secondly the external radius will be bigger than 100.00001m because the cone comes down at an angle. That is, the angle between the base and the side of the cone is not 90 degrees so this gives a trapezoidal shape at the end of the cloth. It is not the end of a rectangle.
These 2 factors together may be enough to account for the difference in volume.

The first answer is definitely the best one.

Now I think I am happy

Much appreciated, thanks for the help