To find out how long the air tank will last at different depths, we need to calculate the total volume of air that the diver breathes per minute and then divide the volume of air in the tank by the rate of air consumption.
Given:
Volume of air tank (V) = 0.010 m^3
Initial pressure (P_1) = 1.0x10^7 Pa
Rate of air consumption (R) = 0.400 L/s
Depth (d) = 1.0 m and 10.0 m
Step 1: Convert the rate of air consumption to m^3/min
1 L = 0.001 m^3
0.400 L/s = 0.400 * (1/60) m^3/min
R = 0.400/60 m^3/min
Step 2: Calculate the final pressure (P_2) at each depth using Boyle's law.
Boyle's law states that P_1 * V_1 = P_2 * V_2, where V_1 is the initial volume and V_2 is the final volume.
V_2 = V_1 + (V_1 / g) * d, where g is the acceleration due to gravity (9.81 m/s^2)
P_2 = (P_1 * V_1) / V_2
For depth of 1.0 m,
V_2 = 0.010 m^3 + (0.010 m^3 / 9.81 m/s^2) * 1.0 m
V_2 = 0.020 m^3
P_2 = (1.0x10^7 Pa * 0.010 m^3) / 0.020 m^3
P_2 = 5.0x10^6 Pa
For depth of 10.0 m,
V_2 = 0.010 m^3 + (0.010 m^3 / 9.81 m/s^2) * 10.0 m
V_2 = 0.110 m^3
P_2 = (1.0x10^7 Pa * 0.010 m^3) / 0.110 m^3
P_2 = 9.1x10^5 Pa
Step 3: Calculate the time (T) the tank will last at each depth using the formula:
T = V / (R * P_2)
For depth of 1.0 m,
T = 0.010 m^3 / (0.400/60 m^3/min * 5.0x10^6 Pa)
T ≈ 72 min
For depth of 10.0 m,
T = 0.010 m^3 / (0.400/60 m^3/min * 9.1x10^5 Pa)
T ≈ 624 min
Therefore, the air tank will last approximately 72 minutes at a depth of 1.0 m and 624 minutes at a depth of 10.0 m.