I am trying to calculate the approximate ABV (alcohol by volume) of a watermelon that I am saturating with vodka.
Details
If I were to cut the watermelon in half length-wise, its face would be an approximate oval (a little rounder, but I mostly need a ball-park) with a length of 12 inches and width of 8 inches. It weighs about 8 pounds. Watermelon.org suggests that a watermelon is 70% flesh and 30% rind. I assume that the rind will still absorb some of the alcohol, but at a diminished rate. For this question, I will assume that the rind absorption rate will be 50% that of the flesh. (I don't know if it is relevant or not, but that website also states that watermelon is 92% water.)
I will be using a 25.3 fl oz (750 ml) bottle of 40% abv vodka. I will assume that 100% of the vodka will be absorbed by the flesh and rind and it will be evenly distributed throughout.
Problem
I know that the equation to calculate the volume for a prolate ellipsoid is 4/3π(1/2*length)((1/2*width)^2), which for my watermelon is 4/3π*6*4*4 ≈ 402 in^3. This is where things get a little hazy and since I would rather not have someone getting alcohol posisioning on my head, I figured it would be time to ask the professional here at Math.SE. I am assuming that 280.4 in^3 (402*.7) would be flesh and 120.6 in^3 (402*.3) would be rind, but I am at a complete loss for how to proceed with calculating ABV from this information.
If any of my assumptions or calculations appear incorrect, please let me know and explain -- after all, I am here to learn.
You can calculate the volume of the watermelon by submersing it in some known volume of water $x$ and then measuring the volume again $y$. The volume of the watermelon is $V=y-x$ (let us suppose we measure volume in mL)
(source: archimedespalimpsest.org)
According to the figures given , $70\%$ is flesh and $30\%$ rind (by weight). If the absorption rate of rind is $50\%$ that of flesh, then the proportion of the total vodka that ends up in the flesh, $f$, satisfies
$$\frac{f}{0.7}=\frac{(1-f)}{0.5(0.3)}$$
so that $f=14/17\approx 0.82$.
If the vodka is proportion $a$ alcohol and the volume of vodka added is $v$, then the percentage of alcohol in the rind of the watermelon (by volume) is:
$$\frac{avf}{0.7V}.$$
So if $a=0.4$, $v=750$mL and $f=0.82$ we get
$$\frac{0.4(750)(0.82)}{0.7V}\approx \frac{351}{V}$$
Your calculations gave a volume $V$ of watermelon of about $6588$mL. The ABV would then be about $0.053$, or $5.3\%$.
If you want an ABV of $w$ for a watermelon of volume $V$, you should add a volume $2.125wV$ of $40\%$ vodka.