Video Transcript
The circular disc preserves its
shape as it shrinks. What is the rate of change of its
area with respect to radius when the radius is 59 centimetres?
We begin by recalling the formula
that allows us to calculate the rate of change of a function at a given point when
𝑥 is equal to 𝑎. It’s 𝑓 prime of 𝑎 equals the
limit as ℎ approaches zero of 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 over ℎ, where 𝑓 prime
is the derivative of the function. But we don’t seem to have a
function here. So let’s consider what we do know
about the area of a circle. It’s given by the formula 𝐴 equals
𝜋𝑟 squared. We could write this as 𝐴 of
𝑟. 𝐴 is a function of 𝑟. That means that the rate of change
of 𝐴 with respect to 𝑟 is the derivative of 𝐴 with respect to 𝑟.
Now, we’re trying to find the rate
of change when the radius is equal to 59. So we’re going to let 𝐴 be equal
to 59. We want to find 𝐴 prime of 59. And by definition, that must be
equal to the limit as ℎ approaches zero of 𝐴 of 59 plus ℎ minus 𝐴 of 59 all over
ℎ. Let’s work out what 𝐴 of 59 plus ℎ
and 𝐴 of 59 actually are. 𝐴 of 𝑟 is 𝜋𝑟 squared. So 𝐴 of 59 plus ℎ is 𝜋 times 59
plus ℎ squared. We distribute our parentheses. And we see that this is equal to 𝜋
times 3481 plus 118ℎ plus ℎ squared. Similarly, 𝐴 of 59 is 𝜋 times 59
squared, which is 3481𝜋. We can replace 𝐴 of 59 plus ℎ and
𝐴 of 59 with these two expressions in our definition for the derivative. And when we factor by 𝜋, we see
the numerator is 𝜋 times 3481 plus 118ℎ plus ℎ squared minus 3481. Now, of course, these give us
zero.
So we’re looking for the limit as ℎ
approaches zero of 𝜋 times 118ℎ plus ℎ squared all over ℎ. And you might now spot we can
actually divide through by ℎ. And our derivative is now the limit
as ℎ approaches zero of 𝜋 times 118 plus ℎ. We’re now ready to perform direct
substitution. We let ℎ be equal to zero. And when we do, we find that 𝐴
prime of 59 equals 118𝜋. The rate of change of the circular
disc’s area with respect to its radius is 118𝜋 centimetres squared per
centimetre. Now, you might be inclined to think
that the answer should be negative. We’re told that the circular disc
is shrinking. However, that’s a bit of a
trick. The area changes in the same
positive or negative direction as the radius. So, in fact, it is indeed a
positive rate of change.
