An exotic spherical jellyfish has a bulk modulus B . Close to the surface of the sea (depth d=0 ), its radius is R . When it dives to a depth d ( d R ), its radius is reduced by R > 0 . Given the density of the incompressible sea water , and the uniform acceleration due to gravity g such that g d B , what is R R ?

Step-by-step solution

The change in pressure at a depth d is given by P = g d . The bulk modulus B is defined as the ratio of volumetric stress to volumetric strain: B = - P V V V V = - g d B The new volume V' of the jellyfish at depth d is: V' = V + V = V (1 - g d B ) Since the jellyfish is spherical, its volume is proportional to the cube of its radius ( V = 4 3 R^3 ). The new radius is R - R , so the ratio of the new volume to the initial volume is: V' V = ( R - R R )^3 = (1 - R R )^3 Equating the two expressions for V' V : (1 - R R