Hard question on science quiz!

Question

This is the question we have been asked:

Gemstones are measured in carats (CD), with a carat having a mass of 200.0 milligrams.

A unit of area used in nuclear physics is the barn (b). One barn equals 1 x 10^-28 m², a typical nuclear cross section.

You probably know the following units of length: one yard (yd)= 3 feet (ft), one foot (ft)= 12 inches (in), one inch = 2.54 cm.

Bugs Bunny wishes to convert the density of water from one gram/cm³ (g/cm^3) into his preferred units, carats/barn·yard (CD/b*yd)

What value should he find? (4 marks)

Answer 1

Wow, what a question!

The hardest part about this question will be keeping track of all the conversions. It might help to start with one conversion (gram to carat, for example) and go one at a time.

So let’s start with grams to carats.

1 gram has 1000 milligrams, so:

1 gram/cm^3 * 1000mg/g =1000mg/cm^3

1 carat has 200.0mg, so:

1000 mg/cm^3 * 1 carat/200mg = 5 carats/cm^3

Now let’s convert to barns. Since our denominator has cm^3 and barns have the same dimensions as cm^2, we’ll split our cm^3 term in our denominator into cm * cm^2 and use the cm^2 term to convert to barns. The remaining cm term will be converted into yards.

So doing cm^2 to barns:

We know that 100 cm are in 1 m

5 carats/(cm^2 * cm) * 100 cm/m = 500 carats/(cm * m * cm).

Now we do this conversion twice, as we want a unit of m^2 to convert to barns:

500 carats/ (cm * m * cm) * 100 cm/m = 50000 carats/(m^2 * cm)

Finally, given that 1 barn is 1 * 10^-28 m^2, we can convert from m^2 to barns:

50000 carats/(m^2 * cm) * 1* 10^-28 m^2 / barn = 5 * 10^-24 carats /(barn * cm)

Whew! That one was tricky and involved some scientific notation wrangling.

The last step is to convert the remaining cm term in the denominator into a yard.

First, we know that 1 inch is 2.54 cm:

5 * 10^-24 carats/(barn * cm) * 2.54 cm/inch = 1.27 * 10^-23 carats/(barn * inch)

And finally, 36 inches in a yard (technically, this is two steps: 12 inches in a foot, 3 feet in a yard, but I think this conversion is commonplace enough that we can fuse the two)：

1.27 * 10^-23 carats/(barn * inch) * 36 inches/ 1 yard = 4.572 * 10 ^-22 carats/(barn * yard)