The atmospheric pressure P in pounds per square inch at an altitude of h feet above the Earth is given by the equation P = 14.7e−0.00005h. Find the pressure at each altitude. (Round your answers to two decimal places.) (a) 0 ft lb/in2 (b) 4000 ft lb/in2 (c) 10,000 ft lb/in2
The Correct Answer and Explanation is:
To solve for the atmospheric pressure at various altitudes, we will use the given formula: P=14.7e−0.00005hP = 14.7e^{-0.00005h}P=14.7e−0.00005h
where:
- PPP is the atmospheric pressure in pounds per square inch (lb/in²),
- hhh is the altitude in feet,
- eee is the base of the natural logarithm (approximately 2.71828).
Step-by-step Calculation:
(a) Pressure at 0 feet:
When the altitude is 0 feet (sea level), we can substitute h=0h = 0h=0 into the equation: P=14.7e−0.00005(0)=14.7e0=14.7 lb/in2P = 14.7e^{-0.00005(0)} = 14.7e^0 = 14.7 \, \text{lb/in}^2P=14.7e−0.00005(0)=14.7e0=14.7lb/in2
Thus, the pressure at 0 feet is 14.7 lb/in².
(b) Pressure at 4000 feet:
Now, let’s substitute h=4000h = 4000h=4000 feet into the equation: P=14.7e−0.00005(4000)=14.7e−0.2P = 14.7e^{-0.00005(4000)} = 14.7e^{-0.2}P=14.7e−0.00005(4000)=14.7e−0.2
Using e−0.2≈0.8187e^{-0.2} \approx 0.8187e−0.2≈0.8187, we get: P=14.7×0.8187≈12.04 lb/in2P = 14.7 \times 0.8187 \approx 12.04 \, \text{lb/in}^2P=14.7×0.8187≈12.04lb/in2
Thus, the pressure at 4000 feet is 12.04 lb/in².
(c) Pressure at 10,000 feet:
Finally, for h=10,000h = 10,000h=10,000 feet, we substitute into the equation: P=14.7e−0.00005(10000)=14.7e−0.5P = 14.7e^{-0.00005(10000)} = 14.7e^{-0.5}P=14.7e−0.00005(10000)=14.7e−0.5
Using e−0.5≈0.6065e^{-0.5} \approx 0.6065e−0.5≈0.6065, we get: P=14.7×0.6065≈8.91 lb/in2P = 14.7 \times 0.6065 \approx 8.91 \, \text{lb/in}^2P=14.7×0.6065≈8.91lb/in2
Thus, the pressure at 10,000 feet is 8.91 lb/in².
Conclusion:
- (a) Pressure at 0 ft: 14.7 lb/in²
- (b) Pressure at 4000 ft: 12.04 lb/in²
- (c) Pressure at 10,000 ft: 8.91 lb/in²
The equation shows that as altitude increases, the atmospheric pressure decreases exponentially due to the e−0.00005he^{-0.00005h}e−0.00005h term. This is consistent with the fact that the higher you go, the less air there is above you, which leads to lower pressure.
