A cylinder has a length of 𝑥
meters and the radius of its base is 𝑟 meters. The length 𝑥 is increased by 20
percent. The radius 𝑟 is decreased by 40
percent. Calculate and describe the
percentage change in the volume of the cylinder.
Remember the formula for volume of
a prism is the area of that prism’s cross section multiplied by its length. The cross section of the cylinder
is a circle. And the formula for area of a
circle with a radius of 𝑟 is 𝜋𝑟 squared.
So the volume of a cylinder is 𝜋
multiplied by its radius squared multiplied by its length. We can use this formula to
calculate the volume of the cylinder before the length and radius have changed.
The area of the cross section is
𝜋𝑟 squared and its length is 𝑥. So the volume of the original
cylinder is 𝜋𝑟 squared multiplied by 𝑥. Its length 𝑥 is then increased by
Since the original length will be
100 percent and we’re increasing it by 20 percent, that means we now have 120
percent of the original length. As a decimal, 120 percent is
equivalent to 1.2. Because to change from a percentage
into a decimal, do we divide by 100. And 120 divided by 100 is 1.2.
This means the new length can be
found by multiplying the original by 1.2; that’s 1.2𝑥.
We perform a similar process to
find the new radius of the cylinder. This time, it’s been decreased by
40 percent. 100 minus 40 is 60. So the new radius is 60 percent the
length of the old radius. 60 percent is equivalent to 0.6
since 60 divided by 100 is 0.6.
This means we can find the radius
of the new cylinder by multiplying the radius of the original cylinder by 0.6. That’s 0.6𝑟. And we can substitute these values
into the formula for the volume.
This time, 𝜋 multiplied by the
radius squared is 𝜋 multiplied by 0.6𝑟 squared. And the length is 1.2𝑥. 0.6𝑟 all squared is the same as
0.6 squared multiplied by 𝑟 squared.
And since multiplication is
commutative, that means we can do it in any order. We can rewrite this slightly as 0.6
squared multiplied by 1.2 squared multiplied by 𝜋𝑟 squared 𝑥. 0.6 squared multiplied by 1.2 is
the same as 0.432.
And we now have expressions for the
volume of the original cylinder and the new cylinder. We’re not done though. We need to calculate and describe
the percentage change in the volume of the cylinder.
The formula we can use for
percentage change is to find the change itself, divide that by the original, and
then multiply it by 100. The change is the new minus the
old; that’s 0.432 𝜋𝑟 squared 𝑥 minus 𝜋𝑟 squared 𝑥.
And we can factorise this
expression. This will make our life easier in a
moment by taking out a common factor of 𝜋𝑟 squared 𝑥. Doing so and we get 𝜋𝑟 squared 𝑥
multiplied by 0.432 minus one. 0.432 minus one is negative
0.568. So the change is negative 0.568
𝜋𝑟 squared 𝑥.
Let’s substitute this into the
formula for percentage change. It’s negative 0.568 𝜋𝑟 squared 𝑥
over the original, which we worked out to be 𝜋𝑟 squared 𝑥. And then, we multiply that by
Notice how both the numerator and
the denominator of this fraction have a common factor of 𝜋𝑟 squared 𝑥. So we’re left with negative 0.568
multiplied by 100, which is negative 56.8 percent.
A negative percentage change means
it’s decreased in volume. This is a 56.8 percent