The atmospheric pressure P in pounds per square inch (psi) is given by
P = 14.7 e−0.21 awhere a is the altitude above sea level (in miles). If a city has an atmospheric pressure of 11.11 psi, what is its altitude? (Recall that 1 mi = 5,280 ft.
just solve for a:
14.7 e^(-0.21a) = 11.11
e^(-0.21a) = 0.7558
-0.21a = ln 07558 = -0.28
a = 4/3
To find the altitude of the city, we need to use the equation P = 14.7 e^(-0.21a), where P is the atmospheric pressure in psi and a is the altitude above sea level in miles.
Since we are given that the atmospheric pressure in the city is 11.11 psi, we can substitute this value into the equation:
11.11 = 14.7 e^(-0.21a)
To solve for a, we will first divide both sides of the equation by 14.7:
11.11/14.7 = e^(-0.21a)
Next, we can take the natural logarithm (ln) of both sides of the equation to get rid of the exponential:
ln(11.11/14.7) = -0.21a
Now, we can solve for a by dividing both sides of the equation by -0.21:
a = ln(11.11/14.7)/(-0.21)
Using a scientific calculator, we can evaluate the right side of the equation to find the value of a:
a ≈ ln(0.7548)/(-0.21)
a ≈ -1.277 miles
Since we are looking for the altitude above sea level, the negative value does not make sense in this context. Therefore, we can conclude that the city's altitude is approximately 1.277 miles above sea level.
To convert this altitude to feet, we can multiply it by the conversion factor 5,280 ft/mi:
Altitude ≈ 1.277 mi * 5,280 ft/mi
Altitude ≈ 6,747.36 ft
Therefore, the city's altitude is approximately 6,747.36 feet above sea level.