Some years ago I presented a radio programme for BBC Radio 4 on the design of the perfect sports bra. One of our problems in preparing the programme was to work out the appropriate cup dimensions, which I was doing surreptitiously while talking to my producer by holding my hand in the appropriate cupped shape and trying to figure the approximate size. She spotted what I was doing and exclaimed “Ah. That’s an SBH.” “A what?” “An SBH. A Standard British Handful.”

Which information came in very useful when I was trying to figure how much wine you could get into a bra. No, it’s not a joke; in fact, it’s a neat little problem in high school geometry. According to the advertisement (see above) there is now a bra designed to hold a full bottle of wine in a gap between the inner and outer layers. It is called (for obvious reasons) the “Wine Rack Bra.” My question is: How wide does the gap need to be?

The surprising answer, assuming that the handful is spherical and has a diameter of 16cm, is that the gap needs only to be a centimeter wide! In a couple of days’ time I’ll put up the workings for those who might still be struggling.

IMAGE: There are multiple sources for this image e.g.

http://www.winerackbraflask.com/WineRack_Bra_Flask_for_Sneaky_Drinking_p/winerack.htm

THE ANSWER:

The volume of the hemispherical inner cup is 2πR^{3}/3, where R is the radius.

The volume of the outer cup is 2π(R+x)^{3}/3, where x is the width of the gap.

The difference (the volume of the gap) is (2π/3){(R+x)^{3} -R^{3}}

But (R+x)^{3} = R^{3 }+ 3R^{2}x + 3Rx^{2} + x^{3 }

≈ R^{3 }+ 3R^{2}x because x is much smaller than R

So the volume of the gap is just (2π/3){3R^{2}x}. And it has to hold a full wine bottle!

A wine bottle holds 750 cubic centimeters. Inverting the equation and putting the numbers in (R = 8cm; R^{2} = 64 cm^{2}), we find that:

x ≈ 750/128π

= 1.86 cm

So why did I say that x is about 1cm? Because, when you look at the picture, it seems that the side and upper panels of the bra are also have wine-holding gaps, of about the same surface area as the cups! It just goes to show that, if you are theorizing in science, it pays to be theorizing about the right facts.