A submarine submerges by admitting seawater \((S=1.03)\) into its ballast tanks. The amount of water admitted is controlled by air pressure, because seawater will cease to flow into the tank when the internal pressure (at the hull penetration) is equal to the hydrostatic pressure at the depth of the submarine. Consider a ballast tank, which can be modeled as a vertical half-cylinder \((R=8 \mathrm{ft}\), \(L=20 \mathrm{ft}\) ) for which the air pressure control valve has failed shut. The failure occurred at the beginning of a dive from \(60 \mathrm{ft}\) to \(1000 \mathrm{ft}\). The tank was initially filled with seawater to a depth of \(2 \mathrm{ft}\) and the air was at a temperature of \(40^{\circ} \mathrm{F}\). As the weight of water in the tank is important in maintaining the boat's attitude, determine the weight of water in the tank as a function of depth during the dive. You may assume that tank internal pressure is always in equilibrium with the ocean's hydrostatic pressure and that the inlet pipe to the tank is at the bottom of the tank and penetrates the hull at the "depth" of the submarine.