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

Kathryn Kingham

The circumference of a circle as a function of its radius is given by 𝐶(𝑟) = 2𝜋𝑟. Express the radius of a circle as a function of its circumference, denoting it by 𝑟(𝐶), and then find 𝑟(36𝜋).

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Video Transcript

The circumference of a circle as a function of its radius is given by 𝐢 of π‘Ÿ equals two πœ‹π‘Ÿ. Express the radius of a circle as a function of its circumference denoting π‘Ÿ of 𝐢 and then find π‘Ÿ if the circumference is 36πœ‹.

Here’s what we know. That finding the circumference in terms of π‘Ÿ equals two πœ‹π‘Ÿ. This formula helps us find the circumference if we’re given the radius. We wanna transform this function into something that looks like this: π‘Ÿ of 𝐢. And that would help us find the radius if we were given the circumference. So where should we start? We wanna start by isolating this variable β€” getting the π‘Ÿ by itself.

Right now, π‘Ÿ is being multiplied by two πœ‹. I can use the multiplicative inverse of two πœ‹ one over two πœ‹ to multiply these values together equals one. In other words, they cancel each other out: two πœ‹ times one over two πœ‹ equals one. But the multiplication property of equality tells me that if I multiply by something on one side of the equation, I have to multiply by the same thing on the other side. So now we have to multiply the circumference by one over two πœ‹.

Now, we have the circumference divided by two πœ‹ equals π‘Ÿ. Because the circumference is no longer a function of the radius anymore, we’re going to just write 𝐢 to represent the circumference. And because the radius is now a function of the circumference, we can write π‘Ÿ of 𝐢. Just for clarity and because that’s how we usually write a function, I’ll move that π‘Ÿ of 𝐢 to the left and say the radius as a function of the circumference equals the circumference divided by two πœ‹; that’s here.

Our next step will be to solve for the radius if we’re given a circumference of 36πœ‹. So we’ve plugged in 36πœ‹ , where our circumference 𝐢 variable was 36 divided by two equals 18 and πœ‹ divided by πœ‹ equals one. The radius of a circle with a circumference 36πœ‹ equals 18. And here is our reordered formula that helped us find that.