Question

Two balls are eight inches and four inches in diameter, respectively. How many times “bigger” is the larger? Explain how you know using your work.

Answer 1

The ball with diameter 8 ( radius 4) is
8 times bigger
than the ball with diameter 4 (radius 2)

Explanation:

We will use volume as an indication of bigger and smaller

For a spherical ball, volume

`V = (4)/(3)\pi r^3\\`

where r is the radius

Since 4/3 and π are constants we can rewrite this equation as
V = k r³

where k = 4/3 x π

We see that the greater the radius, the greater the volume

First note that diameter = r/2
So ball with diameter 8 has radius 8/2 = 4

and ball with diameter 4 has radius 4/2 = 2

Volume of ball with radius 4 = V₄ = k x 4³ = 64k

Volume of ball with radius 2 = V₂ = k x 2³ = 8k

V₄/V₂ =64k/8k = 8

So the ball with radius 4 (diameter 8) is 8 times bigger than the ball with radius 2(diameter 4)

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Another way to understand this is that the diameter of the larger ball is 2 x the diameter of the smaller ball.

Hence the radius of the larger ball is also 2 x the radius of the smaller ball

Therefore larger to smaller volume = 2³ =8