Question

If the radius of the earth is roughly 3,960 miles, how many times lager is the volume of the earth than the volume of the ping pong ball? A ping pong ball has a radius of 0.7441 inches. 1 miles = 5,280

Answer 1

The volume of a sphere is V = (4/3)πr^3

V(earth)/V(ping pong ball)
= [(4/3)*π*(re)^3] / [(4/3)*π*(re)^3]
= (re)^3/(rp)^3
where re = radius of the earth, rp = radius of the ping pong ball

You need to use a calculator because we expect that this is a large or very small number.

Given that re = 3960 miles and rp = 0.7441 inches (1 miles/5280 feet)(1 feet/12 inches) = 1.1744 x 10^(-5) miles

Therefore,
(3960^3) / [1.1744 x 10^(-4)] ^3
= 3.8339 x 10^25 times larger than the ping-pong ball

Answer 2

The volume of a sphere is (4/3) (pi) (radius³) .

The important part is that little ( ³ ) on the radius.
It shows that the volume is proportional to the cube of the radius.
So if we want to compare the volumes of two spheres, all we need
to do is find the ratio of their radii (radiuses) and cube it.

Of course, the two radii have to be in the same units.
At the moment, I'd rather change the radius of the Earth to inches
than change the radius of the pingpong ball to miles.

For the Earth ...

(3960 miles) x (5280 feet/mile) x (12 inches/foot)

= (3960 x 5280 x 12) inches

= 250,905,600 inches

The ratio of the radii of the 2 spheres is

250,905,600 / 0.7441

= 337,193,388 . (It actually comes out even !)

The cube of that ugly number is the ratio of their volumes.

(337,193,388)³ = 38,338,679.5 billion billion (rounded)