Video: Finding a Height of a Right Circular Cone with a Given Volume and Base

The volume, 𝑉, of a right circular cone with radius length 𝑟 is given by 𝑉 = (1/3) 𝜋𝑟²ℎ. Find the height of a right circular cone with volume 4,312 cm³ and base diameter length 28 cm. (Take 𝜋 = 22/7).

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Video Transcript

The volume, 𝑉, of a right circular cone with radius length 𝑟 is given by 𝑉 is equal to one-third 𝜋𝑟 squared ℎ. Find the height of a right circular cone with volume 4,312 cubic centimeters and base diameter length 28 centimeters. Take 𝜋 equal to 22 over seven.

We are told in the question that the base diameter of the cone is 28 centimeters. The radius is equal to half of the diameter. As a half of 28 is equal to 14, the base radius equals 14 centimeters. We are also told that the volume of the cone is 4,312 cubic centimeters. In this question, we will take 𝜋 as 22 over seven.

We now have two options. We could rearrange the formula for the volume of a cone to make ℎ the subject. We would then be able to substitute in our three values. Alternatively, we can substitute the values first.

Substituting in the values gives us 4,312 is equal to one-third multiplied by 22 over seven multiplied by 14 squared multiplied by ℎ. There are lots of ways of solving this equation to calculate the value of ℎ. We could multiply one-third by 22 over seven by 14 squared. This gives us 616 over three. 4,312 is equal to 616 over three multiplied by ℎ. We could then divide both sides of this equation by 616 over three. 616 divides into 4,312 seven times. Multiplying this by three gives us 21. Therefore, ℎ is equal to 21. The height of a cone with volume 4,312 cubic centimeters and diameter of length 28 centimeters is 21 centimeters.

As previously mentioned, we could’ve rearranged our formula to make ℎ the subject first. We can do this using our knowledge of inverse operations and the balancing method. Firstly, we can multiply both sides of the equation by three. This gives us three 𝑉 is equal to 𝜋𝑟 squared ℎ. We can then divide both sides of this equation by 𝜋𝑟 squared. This gives us ℎ is equal to three 𝑉 divided by 𝜋𝑟 squared. We could then substitute our value of 𝑉, 4,312; value of 𝜋, 22 over seven; and our value of 𝑟, the base radius, 14. ℎ is equal to three multiplied by 4,312 divided by 22 over seven multiplied by 14 squared. Typing this into the calculator once again gives us an answer of ℎ equals 21. The height of the cone is 21 centimeters.

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