Question

If the fractional compression of water at the bottom of an ocean is \(1.5\times10^{-2}\), then the depth of the ocean is \[ \left( B=2.2\times10^9\,\mathrm{N\,m^{-2}}, \; g=10\,\mathrm{m\,s^{-2}} \right) \]

• A. \(3.3\,\mathrm{km}\)
• B. \(1.7\,\mathrm{km}\)
• C. \(1.1\,\mathrm{km}\)
• D. \(2.4\,\mathrm{km}\)

Hint:

Useful relations: \[ \boxed{ B=\frac{\Delta P}{\Delta V/V} } \] and \[ \boxed{ \Delta P=\rho gh. } \]

Answer 1

The Correct Option is A

Step 1: Use the definition of bulk modulus. \[ B = \frac{\Delta P}{\Delta V/V}. \] Given, \[ \frac{\Delta V}{V} = 1.5\times10^{-2}. \] Hence, \[ \Delta P = B\left(\frac{\Delta V}{V}\right) = 2.2\times10^9 \times 1.5\times10^{-2} = 3.3\times10^7\,\mathrm{Pa}. \]

Step 2: Relate pressure to depth. Hydrostatic pressure is \[ \Delta P=\rho gh. \] Taking \[ \rho=1000\,\mathrm{kg\,m^{-3}}, \] \[ h = \frac{3.3\times10^7} {1000\times10} = 3300\,\mathrm{m}. \] \[ h=3.3\,\mathrm{km}. \] Hence, \[ \boxed{3.3\,\mathrm{km}} \] Therefore, \[ \boxed{(A)} \] is the correct answer.