The rate of change of atmospheric
pressure 𝑝 with respect to altitude ℎ is proportional to 𝑝. Which equation describes this
relation? Is it a) 𝑝 equals 𝑘ℎ, b) d𝑝 by
dℎ equals 𝑝? Is it c) d𝑝 by dℎ equals 𝑘 over
𝑝 or d) d𝑝 by dℎ equals 𝑘 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 “rate 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 ℎ.
Change in 𝑝 over an altitude ℎ can
be written as the derivative of 𝑝 with respect to ℎ. And we also have this phrase “is
proportional to.” Let’s say 𝑦 is proportional to
𝑥. If this is the case, we can say
that 𝑦 is equal to some value times 𝑥. We call this some value 𝑘 the
constant of proportionality.
Now in our question, the rate of
change of atmospheric pressure 𝑝 with respect to altitude ℎ is proportional to
𝑝. So we can say that d𝑝 by dℎ is
proportional to 𝑝. And this means, in equation form,
we can say that d𝑝 by dℎ equals 𝑘, that constant of proportionality, times 𝑝. And the correct answer is therefore
d d𝑝 by dℎ is equal to 𝑘𝑝.
Now let’s look at what the other
three would have meant. In c, it tells us that d𝑝 by dℎ is
equal to 𝑘 over 𝑝. This would have meant that the rate
of change of 𝑝 with respect to ℎ is inversely proportional to 𝑝. a, where it says
𝑝 is equal to 𝑘 times ℎ, would have been 𝑝 is proportional to ℎ. 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.