Video Transcript
A spherical balloon leaks helium at
a rate of 48 centimeters cubed per second. What is the rate of change of its
surface area when its radius is 41 centimeters?
The first thing to identify about
this question is that it’s a related rates problem. This means a problem in which two
related quantities are changing over time. We have been given a spherical
balloon. Recall that one of the defining
characteristics of a sphere is its radius, that is, the distance from the center of
the sphere to its circumference. We can mark this on our balloon
diagram. Let’s now examine our question in
more detail. The first piece of information we
are given is the rate at which helium is being lost in centimeters cubed per
second. Here, we are being told the volume
of helium that is being lost per unit of time.
Next, we are asked for the rate of
change of the surface area when the radius takes a given value. More specifically, this means the
rate of change of area per unit of time. The first tools that we will need
for this question are the formulas for the volume and surface area of a sphere with
respect to its radius. These should be familiar to
you. The volume of a sphere which we’ll
represent as 𝑉 is equal to four-thirds 𝜋𝑟 cubed. The surface area of a sphere which
we’ll represent as 𝐴 is equal to four 𝜋𝑟 squared.
Okay, so our question has asked us
to find the rate of change of the surface area with respect to time. We can express this as d𝐴 by
d𝑡. Unfortunately, the formula that we
have for volume is in terms of the radius of the sphere instead of time. This means we cannot directly
differentiate our formula and reach the desired result. Instead, to make progress, we’ll
use an application of the chain rule to write an equivalent statement. That is, d𝐴 by d𝑡 is equal to d𝐴
by d𝑟 multiplied by d𝑟 by d𝑡. Here, we can see the first of these
terms is the derivative of the surface area with respect to the radius. Since we have a formula for surface
area in terms of radius, we can use differentiation to find this term.
Differentiating four 𝜋𝑟 squared
with respect to 𝑟, we get eight 𝜋𝑟. This expression can be substituted
back into our equation for d𝐴 by d𝑡. After substituting this back into
our equation, we are still stuck with this d𝑟 by d𝑡 term. This represents the rate of change
of the radius with respect to time, but we currently don’t have an expression for
that. Right now, we have formulas for the
volume and surface area in terms of the radius. In order to move forward, we’ll
need to use these to express d𝑟 by d𝑡 in a more useful way. To do so, let’s go back to the
first piece of information given to us in the question.
The balloon is leaking helium at a
rate of 48 centimeters cubed per second. Here, we are being given the rate
of change of the volume with respect to time, that is, d𝑉 by d𝑡. The question tells us that the
balloon is losing helium. We therefore say that our rate of
change is negative 48 centimeters cubed per second. It is also worth noting here that
as time increases, our balloon is always decreasing in volume. In other words, it is
shrinking. Logically, since the shape remains
a sphere, we should expect that the surface area of our shrinking balloon will also
be decreasing over time. When we finally get to the answer
our question, we should therefore see a negative number to represent that the
surface area is decreasing over time when the radius of the balloon is 41
centimeters.
Another quick side note, let’s
simply remember that consistently throughout the problem, the units of length in the
system are centimeters and the units of time are seconds. This allows us to set aside the
units during our calculations and add them back at the end of the question. We now have that d𝑉 by d𝑡 is
equal to negative 48, but this does not yet help us with our goal of reexpressing
d𝑟 by d𝑡. Instead, we can return to the
previous trick that we used involving our friend, the chain rule. This allows us to construct an
equivalent statement that d𝑉 by d𝑡 is equal to d𝑉 by d𝑟 multiplied by d𝑟 by
d𝑡. Since these two statements are
equivalent, we can also say this is equal to negative 48.
Looking back at our original
formulas, we have an equation for 𝑉 in terms of 𝑟. This means that we can make
progress by finding d𝑉 by d𝑟. Using differentiation and a little
simplification, we find d𝑉 by d𝑟 is equal to four 𝜋𝑟 squared. Let’s substitute this result back
into the equation. Now dividing both sides of the
equation by four 𝜋𝑟 squared allows us to isolate d𝑟 by d𝑡 on the left-hand
side. This is now an equation expressing
d𝑟 by d𝑡 in terms of the radius of the sphere.
It’s worth mentioning that we have
to be careful since our equation now involves a division by the variable 𝑟. In this case, our equation requires
us to divide by zero when 𝑟 is equal to zero. And hence, d𝑟 by d𝑡 is undefined
at this point. Fortunately, we know the balloon is
shrinking, and we’ll be evaluating the system when the radius is 41 centimeters. This means the undefined point when
the radius is zero does not affect us. Now, we can simplify the right-hand
side of the equation to find that d𝑟 by d𝑡 is equal to negative 12 over 𝜋𝑟
squared.
Great! Let’s substitute this back into the
equation for d𝐴 by d𝑡. On the right-hand side of this
equation, we now have eight 𝜋𝑟 multiplied by negative 12 over 𝜋𝑟 squared. We can cancel the 𝜋 terms and one
of the 𝑟 terms in the denominator. We also multiply eight by negative
12. This leaves us with the result that
d𝐴 by d𝑡 is equal to negative 96 over 𝑟. We now have a general expression
that allows us to find the rate of change of the surface area of our balloon with
respect to time for any given radius 𝑟. Remember this is not valid for the
case of 𝑟 equals zero as we mentioned earlier.
For our final step, we recall that
the question asked us for d𝐴 by d𝑡 when the radius is 41 centimeters. We are now in a position to
substitute the value of 41 directly into our equation. Since there are no useful
simplifications for the fraction negative 96 over 41, we can leave our answer as it
is. However, we must remember to add
the units for surface area over time, that is, centimeters squared per second.
Great! We have now found that the rate of
change of the surface area of the balloon is negative 96 over 41 centimeters squared
per second when the radius of the balloon is 41 centimeters. This is the answer to our
question. As a final note, the negative
result we see matches our earlier observation that the rate of change of the surface
area should be negative, since the balloon is shrinking.
