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# Preventing barotrauma in deep-sea divers

## Problem

In addition to decompression sickness, deep-sea divers must remain wary of injuries arising purely from mechanical forces that act on them as they ascend or descend significant depths. A common class of injuries, known as barotraumas, occur due to mechanical forces occurring within the body due to pressure gradients that occur inside and outside the body. Depending on the rate at which a diver ascends or descends, the fluid pressure inside their ears and other soft tissues may not have time to equilibrate with the external pressure, resulting in a difference in pressure that can cause mechanical strain in sensitive tissues.
In order to combat this effect, many dive suits, masks, and tanks are pressurized in order to combat the cumulative effects of pressure during the dive. However, because barotrauma can occur at depths as low as $20$ meters, occasionally divers may fall victim because they did not wear adequate equipment for the depths that they attain. Particularly troubling are injuries occurring where most of the diver’s bodysuit was safely pressurized, but one component like a mask was not adequately pressurized, leading to pressure gradients across the body that can cause injury.
A diver has devised four different options for a dive to a depth of $40$ meters over $90$ minutes. For safety, she plans to maintain a fixed, positive buoyancy, and manually swim against their buoyancy in order to travel deeper. Her candidate diving procedures are illustrated in a depth versus time graph in Figure 1.
Note: The density of water is denoted by $\rho$ and the acceleration due to gravity is given by $g$
Figure 1: Four different dive procedures for a diver intending to dive $40$ meters in $90$ minutes.
Which of the following gives the difference in pressure between the surface and at dive depth $D$?