A research biologist requires a sturdy yet floatable plastic sphere to carry out a lake project within a local wildlife refuge. the spheres have a diameter of 51.75mm what is the maximum allowable mass the sphere can be in order to remain afloat? Assume the density of water in the lake is 1.0 g/cm^3 at all times throughout the year

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.