It is given that the atmospheric pressure is,\\\\( P_o = \\rho g H \\\\), where \\\\( \\rho \\\\) is the density of water.Let the depth of the lake be d, the pressure at this depth would be,P = \\\\( P_o + \\rho g d \\\\)Hence, P = \\\\( \\rho g (d+H) \\\\)Since, the surrounding temperature is constant, we can assume that the process takes place isothermally. Therefore, we can apply the Boyle's Law\\\\( P_1 V_1 = P_2 V_2 \\\\)Here, 1 denotes the water at depth d and 2 denotes the surface of water.Hence, \\\\( P_1 = \\rho g (d+H) \\\\)\\\\( P_2 = \\rho g H \\\\)\\\\( V_2 = 8V_1 \\\\) (since, radius is doubled, volume becomes 8 times)Substituting the values, in Boyle's Law, d = 8H - H = 7H