To determine the depth of a vessel submerged in fresh water at a pressure of 9.42 ATM, the formula P = Po + pgh is used, where P is the total pressure, Po is the atmospheric pressure, p is the water density (1000 kg/m^3 for fresh water), and h is the depth. The atmospheric pressure at sea level is approximately 1 ATM, so the pressure due to the water column can be calculated by subtracting this from the total pressure. This results in a pressure of 8.42 ATM exerted by the water column. By rearranging the formula to solve for depth (h), the depth can be calculated using the known values. The discussion emphasizes the importance of understanding the relationship between pressure and depth in fluid mechanics.