Final answer:
To fit 5400 L of air into a spherical tank with a width of 83.0 cm, the air must be compressed to approximately 22.78 atm, assuming initial atmospheric pressure. None of the provided answer choices match this calculated pressure, suggesting a potential error in the provided answer options.
Step-by-step explanation:
To calculate the pressure required to compress 5400 L of air into a spherical air tank of 83.0 cm in diameter, we first need to find the volume of the tank in liters. Since the tank is spherical its volume V can be determined using the formula for the volume of a sphere V = 4/3 * π * r^3, where r is the radius of the sphere.
The radius (r) is half of the width of the tank, so r = 83.0 cm / 2 = 41.5 cm. We then convert the radius to meters by dividing by 100 since there are 100 cm in a meter: r = 0.415 m.
Substituting this value into the volume formula gives:
V = 4/3 * π * (0.415 m)^3 = 0.237 m^3
To convert cubic meters to liters, we use the fact that 1 m^3 = 1000 L, so the volume of the tank in liters is V = 0.237 m^3 * 1000 L/m^3 = 237 L.
Using the ideal gas law, we have P1 * V1 = P2 * V2, where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume. Assuming the initial pressure is atmospheric pressure (1 atm) and the initial volume is the amount of air needed (5400 L), we can solve for the final pressure P2.
P2 = P1 * V1 / V2 = 1 atm * 5400 L / 237 L ≈ 22.78 atm.
However, none of the answer choices given match this result. If you have typed the options correctly, then it's possible that there may be a typo or mistake in the question or answer choices. Otherwise, please recheck the question and available answer choices.