A radius of a sphere is to be
three centimeters with a possible error of 0.02 centimeters.
We want to use differentials to approximate
the maximum error in calculating the volume.
So, this change change in volume is called propagated error.
And we can use differential V
to approximate delta V, the actual change in volume.
So using the volume formula for a sphere,
volume equals 4/3 pi r cubed,
differential V is going to be equal
to the derivative of the volume function
with respects to r times differential r.
So looking at the given information again,
the radius will equal three centimeters.
And the change in r, or differential r,
is going to be equal to the possible error
of plus or minus 0.02 centimeters.
So the only thing left to do here
is determine the derivative of the volume function
with respects to r.
So V prime of r is going to be equal to
four pi r squared.
So the propagated error, or delta V,
is going to be approximately equal to differential V,
which is equal to the derivative of V
evaluated at r equals three
times differential r,
which is plus or minus 0.02 centimeters.
Now we'll substitute three in for r
into the derivative function
and then multiply it by plus or minus 0.02.
Now let's go to the calculator.
Four times pi times three squared
times 0.02.
So, because of the error in measuring the radius,
the volume could be off
by a plus or minus 2.26 cubic centimeters.
Now, sometimes this is hard to determine
whether this amount of error is acceptable or not.
And so often you'll also be asked to determine
the relative error, which compares the propagated error
to what the volume should be
if r was actually equal to three.
And it's normally expressed as a percentage.
So our relative error
is going to be equal to differential V,
which we already found as
plus or minus 2.26 divided by the volume of the sphere
if r was equal to exactly three.
So we'd have 4/3 times pi times r cubed.
Notice how one factor of three is going to simplify out here.
So we'll have plus or minus 2.26 divided by,
this is going to be four times pi times nine,
that's 36 pi.
Let's go to the calculator.
2.26 divided by 36 pi.
So it's approximately plus or minus 0.02,
which would be plus or minus 2%.
So notice, when expressed as a percentage,
it does give a better feel for how much of an error
we can actually experience with the radius
being off by a plus or minus 0.02 centimeters.
I hope you found this helpful.