You learn something new...

LJ friend auto194419 mentions a bit of folk wisdom, to the effect that a ripe watermelon will float in water, while one that is not ripe will sink. Unfortunately, tubs of water are generally not available at places where watermelons are sold, so the question arises: Given the length of a watermelon and its girth around the middle, how might one calculate whether or not it is less or more dense than water, i.e., whether it is ripe or not?

Google describes the shape of a watermelon to be approximately that of a prolate spheroid. The volume of such a beast is calculated using the relation

V = (4/3)πba²

where π is the ratio of the circumference of a circle to its diameter, approximately equal to 3.14, b is the radius of the major axis (the long one), and a is the radius of the minor axis (the short one). (As an aside, notice that if the major and minor axes are the same, i.e., a = b, the formula for the volume reduces to that for a sphere.)

Given the end-to-end length of the watermelon w (where w = 2b), and its circumference around the middle c (where c = 2πa), we get:

b = w/2, and
a = c/2π

Substituting these values in the equation for the volume, we get:

V = (4/3) x π x (w/2) x (c/2π) x (c/2π)

which reduces to

V = wc²/6π

Since there are 10³ cm³ of water in a liter, and a liter of water weighs a kilogram, dividing the result by 10³ will obtain the maximum number of kilograms a watermelon can weigh and still be less dense than water.

In auto194419's example, the length of the watermelon was 60 cm and the circumference was 91.5 cm, so we get:

V = (60 x 91.5 x 91.5) / 6 x 3.142 = 26.64 x 10³ cm³;

So, if each of those cubic centimeters was made of water, the watermelon would weigh 26.64 kg (recall that 10³ cm³ of water = 1 kg). The fact that the watermelon actually weighed 15.5 kg means it is less dense than water, and therefore ripe, which turned out to be the case.

To be honest, I always cross my fingers when buying a watermelon. I've tried the "thumping" method, but I apparently don't thump enough watermelons to have developed an accurate ear, so for me buying a watermelon has pretty much been a random affair. I look forward to using auto194419's approach the next time I go shopping for a watermelon.

And people say that they have no use for math. Tsk.

Cheers...