Questions: Many people have imagined that if they were to float the top of a flexible snorkel tube out of the water, they would be able to breathe through it while walking underwater. However, they generally do not consider just how much water pressure opposes the expansion of the chest and the inflation of the lungs. Suppose you can barely breathe while lying on the floor with a 390 N weight on your chest. At what depth h could your chest be below the surface of the for you to be able to barely breathe, assuming that your chest wall has a surface area of 0.10 m^2 ? h= cm

Solution Steps

The pressure is force divided by area. In this case, the force is 390 N and the area is 0.10 m². Therefore, the pressure is:

$P = \frac{F}{A} = \frac{390 \text{ N}}{0.10 \text{ m}^2} = 3900 \text{ Pa}$

The pressure due to the water at a depth $h$ is given by $P = \rho g h$, where $\rho$ is the density of water (approximately $1000 \text{ kg/m}^3$) and $g$ is the acceleration due to gravity (approximately $9.8 \text{ m/s}^2$).

We want to find the depth $h$ at which the water pressure is equal to the pressure calculated in Step 1. Therefore, we set the two pressures equal and solve for $h$:

$P = \rho g h$ $3900 \text{ Pa} = (1000 \text{ kg/m}^3)(9.8 \text{ m/s}^2)h$ $h = \frac{3900 \text{ Pa}}{(1000 \text{ kg/m}^3)(9.8 \text{ m/s}^2)} \approx 0.398 \text{ m}$

$h = 0.398 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 39.8 \text{ cm}$

Final Answer: The depth is approximately 39.8 cm.