The atmospheric pressure P in pounds per square inch (psi) is given by the formula below, where a is the altitude above sea level (in miles). If a city has an atmospheric pressure of 13.25 psi, what is its altitude? (Recall that 1 mi = 5,280 ft. Round your answer to the nearest foot.)
P=14.7e^0.21a
_________ ft
2611 ft
To find the altitude of a city given its atmospheric pressure, we can use the formula P = 14.7e^(0.21a), where P represents the atmospheric pressure in psi, and a represents the altitude above sea level in miles.
In this case, we are given that the atmospheric pressure is 13.25 psi. We need to solve for the corresponding altitude, a.
First, let's set up the equation using the given atmospheric pressure:
13.25 = 14.7e^(0.21a)
To solve for a, we need to isolate the variable. Let's divide both sides of the equation by 14.7:
13.25 / 14.7 = e^(0.21a)
Now, we can take the natural logarithm (ln) of both sides to remove the exponential value:
ln(13.25 / 14.7) = 0.21a
To find a, we need to isolate it on one side of the equation. Let's divide both sides by 0.21:
a = ln(13.25 / 14.7) / 0.21
Using a calculator, we can evaluate the right side of the equation:
a ≈ ln(0.899) / 0.21 ≈ -0.107 / 0.21 ≈ -0.51
Now, we know that the altitude is given in miles, but we are asked to round the answer to the nearest foot. To convert miles to feet, we can use the conversion factor: 1 mile = 5280 feet.
a ≈ -0.51 * 5280 ≈ -2692.8
Finally, rounding to the nearest foot, we get an altitude of approximately -2693 feet.
Therefore, the altitude of the city is approximately -2693 feet. Note that a negative altitude indicates that the city is below sea level.