The maximum allowable mass is 580 g.
Step 1. Calculate the volume of the spheres
V = (4/3)πr^3 = (4/3)π × (51.75 mm^3) = 5.8052 × 10^5 mm^3
Step 2. Convert the volume to cubic centimetres
V = 5.8052 × 10^5 mm^3 × (1 cm/10 mm)^3 = 580.52 cm^3
Step 3. Calculate the mass of water displaced
According to Archimedes’ principle, each sphere displaces
580.52 cm^3 water
Mass of water = 580.52 cm^3 water × (1.0 g water/1 cm^3) = 580 g water
If the mass of a sphere is greater than the mass of water displaced, the sphere will sink.
∴ The maximum allowable mass of the sphere is 580 g.
