WEBVTT
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The rate of change of atmospheric pressure π with respect to altitude β is proportional to π.
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Which equation describes this relation?
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Is it a) π equals πβ, b) dπ by dβ equals π?
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Is it c) dπ by dβ equals π over π or d) dπ by dβ equals π times π?
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To answer this question, weβre going to need to recall a couple of definitions.
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The first is the meaning of this phrase βrate of change.β
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When we find the rate of change, weβre looking to find the derivative.
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Here we want the rate of change of atmospheric pressure π with respect to altitude β.
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Change in π over an altitude β can be written as the derivative of π with respect to β.
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And we also have this phrase βis proportional to.β
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Letβs say π¦ is proportional to π₯.
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If this is the case, we can say that π¦ is equal to some value times π₯.
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We call this some value π the constant of proportionality.
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Now in our question, the rate of change of atmospheric pressure π with respect to altitude β is proportional to π.
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So we can say that dπ by dβ is proportional to π.
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And this means, in equation form, we can say that dπ by dβ equals π, that constant of proportionality, times π.
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And the correct answer is therefore d dπ by dβ is equal to ππ.
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Now letβs look at what the other three would have meant.
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In c, it tells us that dπ by dβ is equal to π over π.
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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.
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And for b, weβre missing that constant of proportionality.
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In fact, the assumption here is that the constant of proportionality is one.
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We canβt assume that without any values.
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So the answer is d.