the atmosphere pressure P in pounds per square inch (psi) is given by

P=14.7e upper o.21a where a is the attitude above sea level in miles. If a city has an atmosphere pressure y 13.29 psi what is its altitude. Recall that one mile = 5,280 feet. Round to the nearest feet

I suspect the equation should read

P = 14.7e-0.21a
in order that the pressure decreases with altitude.

For P=13.29
13.29=14.70.21a
e0.21a = 13.29/14.7
take log on both sides
-0.21a = ln(13.29/14.7)
a=-ln(13.29/14.7)/0.21
= 0.4802 miles
= 2535.3 feet (approx.)
= 2535 feet (approx.)

13.29 = 14.7 e^(.21a)

.90408.. = e^(.21a)

by rules of logs
.21a = ln .90408...
.21a = -.0100835..
a = -.48016 miles
or - 2535 ft below sealevel !!!!!!!

Wow, since the lowest point in North America, Death Valley is only about 260 ft below sea level, you city is a mystery.
... or else your equation is bogus.

To find the altitude, we need to solve the equation P = 14.7 * e^(0.21a) for "a" when P = 13.29 psi.

Plugging in the given value, we have:
13.29 = 14.7 * e^(0.21a)

Divide both sides by 14.7:
13.29 / 14.7 = e^(0.21a)

Take the natural logarithm of both sides:
ln(13.29 / 14.7) = 0.21a

Now divide both sides by 0.21 to solve for "a":
a = ln(13.29 / 14.7) / 0.21

Using a calculator, we can calculate this value:
a ≈ ln(0.903061) / 0.21 ≈ -0.102493

The altitude, a, is approximately -0.102493 miles above sea level.

To convert this to feet, we multiply by 5280:
-0.102493 * 5280 ≈ -541.09

Rounding to the nearest foot, the altitude is approximately -541 feet above sea level.

Note: Since the altitude is negative, it means the city is below sea level.

To find the altitude of the city given the atmospheric pressure, we can rearrange the formula and solve for 'a'. Let's start by substituting the given pressure value:

P = 14.7e^(0.21a)
13.29 = 14.7e^(0.21a)

Now, divide both sides of the equation by 14.7:

13.29/14.7 = e^(0.21a)

Next, take the natural logarithm (ln) of both sides to remove the exponential function:

ln(13.29/14.7) = ln(e^(0.21a))

By the properties of logarithms, the natural logarithm and exponential functions cancel each other out:

ln(13.29/14.7) = 0.21a

Now, divide both sides by 0.21:

ln(13.29/14.7) / 0.21 = a

Using a calculator, evaluate the left-hand side:

a ≈ ln(0.902) / 0.21

The result of this calculation will give you the value of 'a' in miles. Finally, to convert it to feet, multiply 'a' by 5280, as one mile is equal to 5280 feet.