Question Video: Solving a Given Formula with Exponents for a Variable in It and Evaluating It at Given Values of Other Variables

The volume of a right circular cone with radius 𝑟 and height ℎ is 𝑉 = 1/3 𝜋𝑟²ℎ. First, write an equation for the radius of a cone with a height of 12 inches as a function of 𝑉. Then, use this to find the radius of the cone to the nearest whole number given that its volume is 50 cubic inches.

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Video Transcript

The volume of a right circular cone with radius 𝑟 and height ℎ is 𝑉 is equal to one-third 𝜋𝑟 squared ℎ. First, write an equation for the radius of a cone with a height of 12 inches as a function of 𝑉. Then, use this to find the radius of the cone to the nearest whole number given that its volume is 50 cubic inches.

We will begin by considering the formula for the volume of a cone, which is equal to one-third multiplied by 𝜋𝑟 squared multiplied by ℎ. We are told in this question that the height is equal to 12 inches. Substituting this into the formula, we have a third multiplied by 𝜋𝑟 squared multiplied by 12. At this stage, we can multiply our two constants one-third and 12. As one-third of 12 is equal to four, 𝑉 is equal to four 𝜋𝑟 squared.

The first part of the question asked us to write an equation for the radius 𝑟 as a function of 𝑉. This means that we need to make 𝑟 the subject of the formula. We will do this using the balancing method and our knowledge of inverse operations. We begin by dividing both sides by four 𝜋. On the right-hand side, the four 𝜋s cancel. So, we are left with 𝑟 squared is equal to 𝑉 over four 𝜋. As square rooting is the opposite or inverse of squaring, we can then square root both sides of this equation. 𝑟 is, therefore, equal to the square root of 𝑉 divided by or over four 𝜋.

The second part of our question asked us to find the radius when the volume is 50 cubic inches. We need to substitute 𝑉 is equal to 50 into this equation. 𝑟 is equal to the square root of 50 divided by four 𝜋. Typing this into our calculator gives us a value of 𝑟 equal to 1.9947 and so on. As we need to round our answer to the nearest whole number, the deciding number is in the tenths column. If this is five or greater, we round up. The radius of the cone, to the nearest whole number, when the volume is 50 cubic inches and the height is 12 inches, is two inches.

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