# Video: AP Calculus AB Exam 1 β’ Section I β’ Part A β’ Question 19

The rate of change of atmospheric pressure π with respect to altitude β is proportional to π. Which equation describes this relation? [A] π = πβ [B] dπ/dβ = π [C] dπ/dβ = π/π [D] dπ/dβ = ππ

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### Video Transcript

The rate of change of atmospheric pressure π with respect to altitude β is proportional to π. Which equation describes this relation? Is it a) π equals πβ, b) dπ by dβ equals π? Is it c) dπ by dβ equals π over π or d) dπ by dβ equals π times π?

To answer this question, weβre going to need to recall a couple of definitions. The first is the meaning of this phrase βrate of change.β When we find the rate of change, weβre looking to find the derivative. Here we want the rate of change of atmospheric pressure π with respect to altitude β.

Change in π over an altitude β can be written as the derivative of π with respect to β. And we also have this phrase βis proportional to.β Letβs say π¦ is proportional to π₯. If this is the case, we can say that π¦ is equal to some value times π₯. We call this some value π the constant of proportionality.

Now in our question, the rate of change of atmospheric pressure π with respect to altitude β is proportional to π. So we can say that dπ by dβ is proportional to π. And this means, in equation form, we can say that dπ by dβ equals π, that constant of proportionality, times π. And the correct answer is therefore d dπ by dβ is equal to ππ.

Now letβs look at what the other three would have meant. In c, it tells us that dπ by dβ is equal to π over π. This would have meant that the rate of change of π with respect to β is inversely proportional to π. a, where it says π is equal to π times β, would have been π is proportional to β. And for b, weβre missing that constant of proportionality. In fact, the assumption here is that the constant of proportionality is one. We canβt assume that without any values. So the answer is d.