Question

the circumference of a circle can be found using the formula c = 2πr. which is an equivalent equation solved for r? 1. r = cπ 2. r = c(2π) 3. r = \frac{c}{2π} 4. r = \frac{2π}{c}

Explanation:

Step1: Start with the formula

The formula for the circumference of a circle is $C = 2\pi r$. We want to solve for $r$.

Step2: Isolate $r$

Divide both sides of the equation $C = 2\pi r$ by $2\pi$. We get $r=\frac{C}{2\pi}$.

Step3: Rearrange to match the form

If we want an equation in terms of $1$, we can rewrite it as $1=\frac{2\pi}{C}\times r$. But if we just want to express the relationship for $r$ - solving the original $C = 2\pi r$ for $r$ gives us the correct form - and if we rewrite it in a non - standard way for the purpose of the options, from $C = 2\pi r$ we can get $r=\frac{C}{2\pi}$ which is equivalent to $1=\frac{C}{2\pi r}$ (when cross - multiplying $r=\frac{C}{2\pi}$).

Answer:

$r=\frac{C}{2\pi}$ which is equivalent to the option where $1 = \frac{C}{2\pi r}$ (assuming the last option $1=\frac{2\pi}{C}$ is a mis - type and should be $1=\frac{C}{2\pi r}$). If we assume the options are exactly as written, there is an error in the problem setup as none of the given options are correct when starting from $C = 2\pi r$ and solving for $r$. If we had to pick the closest conceptually wrong option that shows some manipulation attempt, we note that starting from $C = 2\pi r$ and trying to isolate $r$:
We should have $r=\frac{C}{2\pi}$. If we rewrite it in a non - standard way for the options, from $C = 2\pi r$ we can cross - multiply to get $1=\frac{C}{2\pi r}$ (but this is a non - standard way of writing it compared to normal algebraic manipulations). Among the given options, none are correct as written. If we assume a mis - type in the last option and it was meant to be $1=\frac{C}{2\pi r}$, that would be the closest to the correct manipulation of the formula $C = 2\pi r$.

If we assume the question is just about algebraic manipulation of the formula $C = 2\pi r$ to solve for $r$ and re - arrange in a non - standard form for the options:
The correct steps lead to $r=\frac{C}{2\pi}$ and a non - standard re - arrangement could be written as $1=\frac{C}{2\pi r}$ (if we force it to match an option - like form). But again, as the options are written exactly as given, there is no correct answer. If we had to pick the closest in terms of showing an attempt at manipulation, we note the form related to the correct $r=\frac{C}{2\pi}$ re - arranged in a non - standard way for the options context.

If we assume the last option was meant to be $1=\frac{C}{2\pi r}$ (a likely mis - type), then that would be the "answer" in the sense of being the closest to the correct manipulation of the formula $C = 2\pi r$.

So, if we assume a mis - type in the last option:
The answer is the option that is conceptually closest to $1=\frac{C}{2\pi r}$ (last option with a possible correction). If no mis - type is assumed, there is no correct answer among the given options.