Physics Question on Bouyancy *EASY 10 POINTS*

Hey guys, any help on this question would be a blessing! Steps and explanations are welcome!

bar of gold sinks to the ocean floor at a depth of 3.9km. Assuming that the
temperature of the ocean remains approximately constant with depth and that the
density of sea water is 1050 kg/m3, by what fraction has the gold’s volume contracted?
(ie, find V2/V1, where V1 is the gold’s initial volume and V2 is the gold’s volume on
the ocean floor).

The gold bar will contract as it is subjected to the pressure of the seawater. There is a physical property called compressibility, which is described as (see Source 1):

" . . . compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change."

Compressibility β is defined by this formula:

(1) β = - (1 / V) * (ΔV / Δp),

where:
V = the original (unpressurized) volume
ΔV = the decrease in V resulting from
Δp = the increase in pressure

The negative sign is there to make β positive, since ΔV / Δp is negative.

We are asked to find V2/V1, where V1 is the gold’s initial volume and V2 is the gold’s volume on
the ocean floor.

So we need to find an equation for V2 using β and ΔV and Δp. The definition of β helps us, because it is the relative volume change in response to a pressure change. So:

(2) V2 = V1 + ΔV (remember that ΔV is negative)

and solving (1) for ΔV:

(3) ΔV = - β * V1 * Δp

Substituting (3) into (2):

(4) V2 = V1 + ΔV

= V1 - β * V1 * Δp

= V1 * (1 - β * Δp)


Therefore,

(5) V2/V1 = [V1 * (1 - β * Δp) ] / V1

= (1 - β * Δp)


Compressibility β is defined to be the reciprocal of the bulk modulus K (see Source 2), and we can find the bulk modulus for gold

KAu = 180GPa, so