Video: Calculating the Radius of a Cone given Its Height and Volume

A cone has a perpendicular height of 92 inches and a volume of 420𝜋 cubic inches. Work out the radius of the cone, giving your answer to the nearest inch.

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Video Transcript

A cone has a perpendicular height of 92 inches and a volume of 420𝜋 cubic inches. Work out the radius of the cone, giving your answer to the nearest inch.

In order to answer this question, we need to recall the formula for the volume of a cone. This is equal to one-third 𝜋𝑟 squared multiplied by ℎ, where ℎ is the perpendicular height of the cone. In this question, we know that the volume is 420𝜋 cubic inches. The height of the cone is 92 inches. This means that 420𝜋 is equal to one-third 𝜋𝑟 squared multiplied by 92. We can divide both sides of this equation by 𝜋. We can also multiply both sides by three. This gives us 1260 is equal to 92𝑟 squared. Next, we can divide both sides by 92 so that 𝑟 squared is equal to 1260 over 92. Finally, we square root both sides of the equation. The radius of the cone is equal to the square root of 1260 over 92. Typing this into the calculator gives us 3.70076 and so on.

As we want the answer to the nearest inch, we need to consider the digit in the tenths column. This is greater than five. So, we round up. And 𝑟 is equal to four. The radius of a cone with perpendicular height 92 inches and a volume of 420𝜋 cubic inches is four inches to the nearest inch. We could check this answer by substituting our value for 𝑟 back into the initial formula.

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