Question Video: Calculating the Density of a Cone | Nagwa Question Video: Calculating the Density of a Cone | Nagwa

Question Video: Calculating the Density of a Cone

A cone with a perpendicular height of 9 feet has a density of 6 lb per cubic foot and a mass of 160 lb. Work out, to two decimal places, the radius of the cone, knowing density = mass/volume.

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Video Transcript

A cone with a perpendicular height of nine feet has a density of six pounds per cubic foot and a mass of 160 pounds. Work out, to two decimal places, the radius of the cone, knowing density is equal to mass divided by volume.

Rearranging the formula density equals mass divided by volume, the volume is equal to the mass divided by the density. We are told that the density of the cone is six pounds per cubic foot. The mass of the cone is 160 pounds. To calculate the volume, we need to divide the mass by the density. As six does not divide exactly into 160, we will simplify the fraction 160 over six. Dividing the numerator and denominator by two gives us 80 over three. The volume of the cone is 80 over three or eighty-thirds cubic feet.

We recall that the volume of any cone is equal to one-third 𝜋𝑟 squared ℎ. In this question, we are told that the perpendicular height, ℎ, is equal to nine feet. Substituting in our values for volume and height gives us 80 over three is equal to one-third 𝜋𝑟 squared multiplied by nine. We can multiply both sides of this equation by three. This gives us 80 is equal to nine 𝜋𝑟 squared.

Dividing both sides by nine 𝜋 gives us 𝑟 squared is equal to 80 over nine 𝜋. We can then square root both sides such that 𝑟 is equal to the square root of 80 over nine 𝜋. Typing this into the calculator gives us 1.6820 and so on. Rounding to two decimal places, we have a radius of 1.68 feet.

A cone with perpendicular height of nine feet, density of six pounds per cubic foot, and mass of 160 pounds will have a radius to two decimal places of 1.68 feet.

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