The atmospheric pressure P in pounds per square inch (psi) is given by

P = 14.7 e−0.21 a
where a is the altitude above sea level (in miles). If a city has an atmospheric pressure of 12.21 psi, what is its altitude? (Recall that 1 mi = 5,280 ft. Round your answer to the nearest foot.)

P(a)=12.21=14.7*e-0.21a

Solve for a:

e-0.21a=12.21/14.7
Take log on both sides, log(e(x))=x
-0.21a=log(12.21/14.7)
a=log(12.21/14.7)/(-0.21)
=?

To find the altitude of the city given its atmospheric pressure, we need to rearrange the formula and solve for "a".

The formula for atmospheric pressure is given as:
P = 14.7e^(-0.21a)

Given that P = 12.21 psi, we can substitute it into the equation:
12.21 = 14.7e^(-0.21a)

To solve for "a", we need to isolate the variable. First, divide both sides of the equation by 14.7:
12.21/14.7 = e^(-0.21a)

Next, take the natural logarithm (ln) of both sides to remove the exponential term:
ln(12.21/14.7) = -0.21a

Now, divide both sides by -0.21 to solve for "a":
a = ln(12.21/14.7)/-0.21

Using a calculator, we can simplify the equation and find the value of "a".

Alternatively, we can use online tools or scientific calculators that support natural logarithm calculations to find the value of a directly.

Let's perform the calculations using a calculator or an online tool.

Using a calculator, we find:
a ≈ 1.6625 miles

Now, convert the altitude from miles to feet. We know that 1 mile is equal to 5,280 feet, so we can multiply the altitude by 5,280 to get the answer in feet:

a (ft) = 1.6625 mi * 5,280 ft/mi

Calculating this, we find:
a ≈ 8,766 ft

Therefore, the altitude of the city is approximately 8,766 feet above sea level.