The circumference of a circle as a function of its radius is given by 𝐶 of 𝑟 equals two 𝜋𝑟. Express the radius of a circle as a function of its circumference denoting 𝑟 of 𝐶 and then find 𝑟 if the circumference is 36𝜋.
Here’s what we know. That finding the circumference in terms of 𝑟 equals two 𝜋𝑟. This formula helps us find the circumference if we’re given the radius. We wanna transform this function into something that looks like this: 𝑟 of 𝐶. 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 — getting the 𝑟 by itself.
Right now, 𝑟 is being multiplied by two 𝜋. I can use the multiplicative inverse of two 𝜋 one over two 𝜋 to multiply these values together equals one. In other words, they cancel each other out: two 𝜋 times one over two 𝜋 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.
Now, we have the circumference divided by two 𝜋 equals 𝑟. Because the circumference is no longer a function of the radius anymore, we’re going to just write 𝐶 to represent the circumference. And because the radius is now a function of the circumference, we can write 𝑟 of 𝐶. Just for clarity and because that’s how we usually write a function, I’ll move that 𝑟 of 𝐶 to the left and say the radius as a function of the circumference equals the circumference divided by two 𝜋; that’s here.
Our next step will be to solve for the radius if we’re given a circumference of 36𝜋. So we’ve plugged in 36𝜋 , where our circumference 𝐶 variable was 36 divided by two equals 18 and 𝜋 divided by 𝜋 equals one. The radius of a circle with a circumference 36𝜋 equals 18. And here is our reordered formula that helped us find that.