Question

If the temperature is constant, then the atmospheric pressure p (in pounds per square inch) varies with the altitude above sea level h in accordance with the law p=p0e^−kh where p0 is the atmospheric pressure at sea level and k is a constant. If the atmospheric pressure is 14.7 lb/in.² at sea level and 13 lb/in.² at 4,000 ft, find the atmospheric pressure at an altitude of 14,000 ft. (Round your answer to two decimal places.)

a) 5.68 lb/in.²
b) 4.87 lb/in.²
c) 4.16 lb/in.²
d) 3.55 lb/in.²

Answer 1

To find the atmospheric pressure at an altitude of 14,000 ft when given the pressure at sea level and at 4,000 ft, we can use the equation p = p0e^(-kh). Substituting the known values and solving for the constant k, we can then plug in the value of k to find the atmospheric pressure at 14,000 ft.

Step-by-step explanation:

The atmospheric pressure at sea level is 14.7 lb/in.², and the pressure at 4,000 ft is 13 lb/in.². To find the atmospheric pressure at an altitude of 14,000 ft, we can use the equation given: p = p0e^(-kh), where p is the atmospheric pressure, p0 is the pressure at sea level, k is a constant, and h is the altitude above sea level.

Substituting the known values, we have 13 = 14.7e^(-k(4000)). To find k, we can rearrange the equation as: e^(-k(4000)) = 13/14.7, and take the natural logarithm of both sides. This gives us -k(4000) = ln(13/14.7), and solving for k, we get k = -ln(13/14.7)/4000.

Now we can use the value of k to find the atmospheric pressure at 14,000 ft: p = 14.7e^(-(-ln(13/14.7)/4000)(14000)).

Calculating this expression, we find that the atmospheric pressure at an altitude of 14,000 ft is approximately 4.16 lb/in.².