Question Video: Rearranging a Given Formula to Highlight a Quantity of Interest

The volume 𝑉 of a right circular cone in terms of its height β„Ž and base radius π‘Ÿ is 𝑉 = (1/3) πœ‹π‘ŸΒ² β„Ž. Give a formula for the radius π‘Ÿ in terms of 𝑉 and β„Ž.

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Video Transcript

The volume 𝑉 of a right circular cone in terms of its height β„Ž and base radius π‘Ÿ is 𝑉 is equal to one-third πœ‹π‘Ÿ squared β„Ž. Give a formula for the radius π‘Ÿ in terms of 𝑉 and β„Ž.

We’re given the formula for the volume of a cone, 𝑉 is equal to one-third πœ‹π‘Ÿ squared β„Ž. In order to give a formula for the radius π‘Ÿ in terms of 𝑉 and β„Ž, we need to rearrange the formula to make π‘Ÿ the subject. We can do this using our knowledge of inverse operations and the balancing method. We can begin by multiplying both sides of the original formula by three. The left-hand side becomes three 𝑉. And as one-third multiplied by three is one, the right-hand side becomes πœ‹π‘Ÿ squared β„Ž.

At this stage, the πœ‹π‘Ÿ squared and β„Ž are all multiplying each other. The opposite or inverse of multiplying is dividing. Our next step is therefore to divide both sides of this new equation by πœ‹ and β„Ž. On the right-hand side, both of these terms cancel. We are therefore left with three 𝑉 over πœ‹β„Ž is equal to π‘Ÿ squared. The opposite or inverse of squaring is square rooting, so we need to square root both sides.

On the right-hand side, this gives us π‘Ÿ as the square root of π‘Ÿ squared is π‘Ÿ. π‘Ÿ is equal to the square root of three 𝑉 over πœ‹β„Ž. π‘Ÿ is the base radius of a cone. Therefore, it must have a positive value. π‘Ÿ is the positive answer of the square root of three 𝑉 over πœ‹β„Ž.

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