The volume of right circular cylinder A is 27 cubic centimeters. What is the volume, in cubic centimeters, of a right circular cylinder with three times the radius and one-third the height of cylinder A?
Let’s firstly consider cylinder A with a radius of 𝑟 and a height of ℎ. The formula for the volume of a cylinder is 𝜋𝑟 squared ℎ. Multiplying 𝜋 by the radius squared by the height for cylinder A gives us an answer of 27 cubic centimeters. We are told that our second cylinder has three times the radius and one-third the height. This means that the radius will be three 𝑟, as 𝑟 multiplied by three is three 𝑟. The height of the cylinder will be one-third ℎ, or ℎ divided by three. We can then substitute these values into the volume formula.
The volume is equal to 𝜋 multiplied by three 𝑟 squared multiplied by one-third ℎ. Three 𝑟 squared is the same as three 𝑟 multiplied by three 𝑟. This is equal to nine 𝑟 squared. The volume simplifies to 𝜋 multiplied by nine 𝑟 squared multiplied by one-third ℎ. Nine multiplied by a third, or a third of nine, is equal to three. This means that nine 𝑟 squared multiplied by one-third ℎ is equal to three 𝑟 squared ℎ. The volume of the new cylinder, in its simplest form, is three 𝜋𝑟 squared ℎ.
We know that 𝜋𝑟 squared ℎ was the volume of cylinder A. This was equal to 27 cubic centimeters. We can, therefore, calculate the volume of the new cylinder by multiplying three by 27. This is equal to 81, as three multiplied by 20 is equal to 60, and three multiplied by seven is 21. The volume of a cylinder with three times the radius and one-third the height of cylinder A is 81 cubic centimeters. This is because the volume is three times as large as cylinder A.