Question Video: Using Inverse Variation to Find an Unknown Mathematics

The height of a right circular cylinder ℎ varies inversely with the square of its radius 𝑟. If ℎ = 93 cm when 𝑟 = 7.5 cm, determine ℎ when 𝑟 = 1.5 cm.

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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.

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