Answer:
about 33.3%
Step-by-step explanation:
The fraction of space not occupied will be the same as the unoccupied fraction of a cylinder of the same height and diameter as a sphere.
<h3>Volume of a cylinder</h3>
The volume of a cylinder of diameter d and height d will be ...
V = πr²h = π(d/2)²(d) = (π/4)d³
<h3>Volume of a sphere</h3>
The volume of a sphere of diameter d is ...
V = 4/3πr³ = 4/3π(d/2)³ = (π/6)d³
<h3>Unoccupied space</h3>
The fraction of space that is occupied is ...
occupied space = (sphere volume)/(cylinder volume)
occupied space = ((π/6)d³) / ((π/4)d³) = 4/6 = 2/3
The fraction of unoccupied space is ...
unoccupied fraction = 1 -2/3 = 1/3
unoccupied fraction ≈ 33.3%
