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

The rate of change of atmospheric pressure 𝑝 with respect to altitude β„Ž is proportional to 𝑝. Which equation describes this relation? [A] 𝑝 = π‘˜β„Ž [B] d𝑝/dβ„Ž = 𝑝 [C] d𝑝/dβ„Ž = π‘˜/𝑝 [D] d𝑝/dβ„Ž = π‘˜π‘

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Video Transcript

The rate of change of atmospheric pressure 𝑝 with respect to altitude β„Ž is proportional to 𝑝. Which equation describes this relation? Is it a) 𝑝 equals π‘˜β„Ž, b) d𝑝 by dβ„Ž equals 𝑝? Is it c) d𝑝 by dβ„Ž equals π‘˜ over 𝑝 or d) d𝑝 by dβ„Ž equals π‘˜ times 𝑝?

To answer this question, we’re going to need to recall a couple of definitions. The first is the meaning of this phrase β€œrate of change.” When we find the rate of change, we’re looking to find the derivative. Here we want the rate of change of atmospheric pressure 𝑝 with respect to altitude β„Ž.

Change in 𝑝 over an altitude β„Ž can be written as the derivative of 𝑝 with respect to β„Ž. And we also have this phrase β€œis proportional to.” Let’s say 𝑦 is proportional to π‘₯. If this is the case, we can say that 𝑦 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 𝑝 with respect to altitude β„Ž is proportional to 𝑝. So we can say that d𝑝 by dβ„Ž is proportional to 𝑝. And this means, in equation form, we can say that d𝑝 by dβ„Ž equals π‘˜, that constant of proportionality, times 𝑝. And the correct answer is therefore d d𝑝 by dβ„Ž is equal to π‘˜π‘.

Now let’s look at what the other three would have meant. In c, it tells us that d𝑝 by dβ„Ž is equal to π‘˜ over 𝑝. This would have meant that the rate of change of 𝑝 with respect to β„Ž is inversely proportional to 𝑝. a, where it says 𝑝 is equal to π‘˜ times β„Ž, would have been 𝑝 is proportional to β„Ž. And for b, we’re missing that constant of proportionality. In fact, the assumption here is that the constant of proportionality is one. We can’t assume that without any values. So the answer is d.

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