Find the height of an aircraft if the atmospheric pressure is 247 millimeters of mercury.

Question

The atmospheric pressure 𝑝 on a balloon or an aircraft decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height ℎ (in kilometers) above sea level by the formula 𝑝 = 760𝑒^−0.145ℎ . Find the height of an aircraft if the atmospheric pressure is 247 millimeters of mercury. (Round to two decimal places.) Show step by step work.

Answer 1

Hi Rashed,

What you have is a formula for pressure as a function of height, p(h). You are given the formula for p(h) and you are given the value for p. All you need to do is plug in this value of p into the formula and then solve for h.

The formula takes on the form p = A*e^(B*h), where A and B are simply constant numbers that are given to us. Without worrying about the numbers, we know we have to find h. In other words, we need to get h by itself.

Let's move A to the left-hand-side of the equation first.

p/A = e^(B*h)

Before we can continue, we see that we have this exponential function e() in our way. A nifty identity that is used often in Algebra is the following:

ln(e^(something)) = something , where ln is the natural logarithm.

In some sense, the ln() and e() 'cancel' each other out!

Knowing this, identity, we can continue rearranging to solve for h. I will leave it to you to take it from here. Remember that if you use a natural log on one side of the equation, you must also use it on the other side of the equation (just like we divided both sides of the equation by A earlier, and not just one side). You should get a positive number for height.

Good luck,

Brad