A solid metal sphere of bulk modulus 1.5 10¹¹ N/m ^2 is lowered into a deep ocean where the pressure is 4.5 10^7 Pa greater than the surface pressure. The percentage decrease in the radius of the sphere is found to be x 10⁻³ % . The value of x is ______.

Step-by-step solution

The bulk modulus B is given by: B = P ( V V ) The fractional decrease in volume is: V V = P B Substitute the given values: V V = 4.5 10^7 1.5 10¹¹ = 3 10⁻⁴ For a sphere, the volume is V = 4 3 R^3 . For small changes, the fractional change in volume is related to the fractional change in radius by: V V = 3 R R Therefore, the fractional decrease in radius is: R R = 1 3 ( V V ) = 1 3 (3 10⁻⁴ ) = 1 10⁻⁴ To find the percentage decrease in radius, multiply by 100 : Percentage decrease = (1 10⁻⁴ ) 100 % = 10⁻² % = 0.01 %