# Month: November 2016

## Sugar Content

This one’s sketchy, but fun. I will assume that Luna has the same density as the average girl. Using the average female’s mass and volume, density can be calculated as $$\rho = \frac{51.7\ kg}{4.99*10^{-2}\ m^3} = 1040\ \frac{kg}{m^3}$$. Recall from “Volume” that Luna’s volume is given by $$V=1.52*10^{23}\ m^3$$. Using density and volume, calculating Luna’s mass is easy.

$$M=\rho V = 1.57*10^{26}\ kg$$

Now a brief non-sequitur. One serving of gummy bears contains 18 grams of sugar. The “World’s Largest Gummy Bear!™” is claimed to contain 51 servings of gummy bears. The seller claims that it weighs “approximately 5 pounds,” but Amazon gives a more exact shipping weight of 4.8 pounds (about 2.2 kilograms). Some dimensional analysis gives an interesting result:

$$\frac{18\ grams\ sugar}{1\ serving} * \frac{51\ servings}{1\ giant\ bear} * \frac{1\ giant\ bear}{2.2\ kg} * \frac{M\ kg}{1\ Luna}$$

$$= 6.6*10^{28}\ grams\ of\ sugar$$

Luna’s sugar content if she were made of gummy bears.

Let’s put this into perspective. The annual world production of sugar in 2015 was 172 million metric tons of sugar, or 1.72*1014 grams. At that rate, it would take $$\frac{6.6*10^{28}\ grams}{1.72*10^{14}\ grams} = 3.85*10^{14}$$ years for the Earth to produce the amount of sugar needed to make a gummy bear the size of Luna.

A number so big is hard to conceptualize, so consider this. At a liberal estimate, the universe is 13.820 billion years old. With some more dimensional analysis, we find:

$$\frac{1\ universe}{1.3820*10^{10}\ years} * \frac{3.85*10^{14}\ years}{1\ Luna} = 28000\ universes$$

Which is the number of universes it would take working in parallel from the beginning of time to create enough sugar to make one Luna by the present day.

## Volume

The average height of a woman living in the United States is 1.622 meters (about 5 feet 3.75 inches). The average weight for a woman of this height ranges from 51.7 to 57.6 kg (about 114 to 127 lbs). To be tactful, I will use the lower bound of weight for calculation. From height and weight, body surface area can be estimated using the Du Bois formula:

$$s_{woman} = 0.007184 * (162.2\ cm)^{0.725} * (51.7\ kg)^{0.425}$$

$$= 1.54\ m^2$$

In 1966, the US Naval Medical Research Institute attempted to create a general formula for body volume using height and weight. In the process, they created a few formulas for more specific groups of people. The most relevant equation, of course, applies to women in the adolescent to adult age group. The study mentions that this formula was developed using test subjects with a weight (kg) to height (cm) ratio between .2 and .8. Luckily, the weight to height ratio of the average woman with $$\frac{51.6\ kg}{162.2\ cm} = 0.318\ \frac{kg}{cm}$$. Since the ratio is within the range, we can use the formula to approximate the volume of the average woman.

$$v_{woman} = 62.90 * (s_{woman}\ m^2) * (\frac{51.7\ kg}{162.2\ cm})^{0.578}$$

$$= 49.9\ liters$$

$$=4.99*10^{-2}\ m^3$$

Now the lunar step. The Lunar Scale Factor cubed gives the ratio of Luna’s volume to the average girl’s volume. Multiplying that by the average girl’s volume gives Luna’s volume.

$$V=\Lambda^3 * (v_{woman}\ m^3)$$

$$=1.52*10^{23}\ m^3$$

For comparison, the Earth’s volume is 1.08321*1012 km3, or 1.08321*1021 m3. By dividing, we find that Luna is $$\frac{V}{1.08321*10^{21}\ m^3}=140$$ times the size of the Earth.