*varied. Today we calculate relative spherical density.*

**density**Here are a few spheres that are NOT sporting equipment. The one on the left is made of solid copper. The one on the right is foam rubber - I squeeze it to give my fingers some relief from too much typing.

Following yesterday's formula, here are the relevant dimensions:

Copper Sphere 5cm diameter 600g weight & Rubber Sphere 7cm diameter 30g weight.

We can compare these two objects by multiplying and dividing their dimensions.

In Excel Math we teach elementary school kids how to demonstrate comparisons - for example:

**Q1. How much larger is the rubber sphere than the copper sphere?**

**A1. We divide 7 by 5 and learn that the rubber sphere is 1.4 times as large ( 7cm ÷ 5cm = 1.4 )**

**Q2. How much larger is the copper sphere than the rubber sphere?**

**A2. It's**

**( 5cm ÷ 7cm = .7 ) or seven-tenths as large.**

**Q3. How much heavier is the copper sphere than the rubber sphere?**

**A3. ( 600g ÷ 30g = 20 ) so the copper sphere is 20 times heavier than the rubber.**

Notice that in all these answers the units disappear. A comparison does not have units!

**Q4. For a given volume, how much heavier is copper than foam rubber?**

**A4. This is much more complex! The formula for volume of a sphere is 4/3 π r³.**

In words, it's four thirds times pi times the radius cubed (multiplied by itself 3 times)

The radius of a sphere is half the diameter so we could end up with something like this:

(4 ÷ 3) x 3.14 x ( d ÷ 2) x ( d ÷ 2) x ( d ÷ 2)

We have to calculate both copper and rubber, so to save time we can do this 4/3π stuff only once and get a "constant value" to use whenever we calculate spherical volume. (4 ÷ 3) x 3.14 = 4.19

Copper

**4.19 x (2.5 x 2.5 x 2.5) = (4.19 x 15.625) = 65 cubic centimeters**

Rubber

**4.19 x (3.5 x 3.5 x 3.5) = (4.19 x 42.875) = 180 cubic centimeters**

Excuse me, what was the question again? Oh yes, relative density (mass or weight per unit of volume)

**Copper**

**600 ÷ 65 = 9.23 grams per cc**

**Rubber**

**30 ÷ 180 = .1666 grams per cc**

Now we can calculate the answer:

**9.23 ÷ .1666 = 55 Copper is 55 times heavier than rubber for the same volume**

**(any volume is 55 times heavier - the units disappear).**