Video Transcript
The height of a right circular
cylinder ℎ varies inversely with the square of its radius 𝑟. If ℎ equals 93 centimeters when 𝑟
equals 7.5 centimeters, determine ℎ when 𝑟 is equal to 1.5 centimeters.
We know that if two variables vary
inversely, as one increases, the other decreases. This means that in this question, ℎ
and 𝑟 squared are inversely proportional. This can be written as ℎ is
proportional to one over 𝑟 squared. This can be rewritten as an
equation using the constant of proportionality 𝑘 such that ℎ is equal to 𝑘 divided
by 𝑟 squared. Multiplying both sides of this
equation by 𝑟 squared gives us ℎ multiplied by 𝑟 squared is equal to 𝑘. When dealing with inverse
proportion or variation, our two variables will have a product equal to some
constant 𝑘.
We are told that when the height of
the cylinder is 93 centimeters, the radius is 7.5 centimeters. This means that we can calculate
the value of 𝑘 by multiplying 93 by 7.5 squared. 7.5 squared is equal to 56.25. Multiplying this by 93 gives us a
value of 𝑘 equal to 5231.25. We can substitute this constant
back into our equation such that ℎ is equal to 5231.25 divided by 𝑟 squared. We want to calculate this value of
ℎ when 𝑟 is equal to 1.5. 1.5 squared is equal to 2.25. This means that ℎ is equal to
5231.25 divided by 2.25. Typing this into the calculator
gives us 2325. The height of the cylinder when the
radius is 1.5 centimeters is 2325 centimeters.
There is a slightly quicker way of
calculating the value of ℎ without working out the constant 𝑘. We begin by considering the fact
that the product of the height and the radius squared must be equal to some constant
𝑘 for any height and radius in this cylinder. This means that in our first
scenario, with a height of 93 centimeters and a radius of 7.5 centimeters, we have
93 multiplied by 7.5 squared. In our second scenario, we have ℎ
multiplied by 1.5 squared as the radius is 1.5 centimeters. We can then divide both sides of
this equation by 1.5 squared. Once again, typing this into the
calculator gives us an answer for ℎ equal to 2325. This confirms that this is the
height of the cylinder when the radius is 1.5 centimeters.
