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
4.19 x (2.5 x 2.5 x 2.5) = (4.19 x 15.625) = 65 cubic centimeters
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)
600 ÷ 65 = 9.23 grams per cc
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).