Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

A wire of length x is bent into the shape of a circle.

Part (a): Express the circumference C of the circle as a function of x.

Part (b): Express the area A of the circle as a function of x.

Step-by-Step Solution

Answer

Part (a): Circumference of the circle as a function is $C\left(x\right)=x$.

Part (b): Area of the circle as a function is $\mathrm{A}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2}{4\mathrm{\pi }}$.

Part (a) Step 1. Given information.

Consider the given question,

Length of the wire x is equal to the circumference of the circle which is $2\mathrm{\pi r}$. Then,

$$\begin{gathered} 2\mathrm{\pi r}=\mathrm{x} \\ \mathrm{r}=\frac{\mathrm{x}}{2\mathrm{\pi }} \end{gathered}$$

Also, circumference of the circle$\left(x\right)=2\mathrm{\pi }\left(\frac{\mathrm{x}}{2\mathrm{\pi }}\right)$

Therefore, circumference as a function of x is $C\left(x\right)=x$.

Part (b) Step 1. Given information.

Consider the given question,
$$\begin{gathered} A\left(x\right)=\mathrm{\pi }{\left(\frac{\mathrm{x}}{2\mathrm{\pi }}\right)}^2 \\ \mathrm{A}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2}{4\mathrm{\pi }} \end{gathered}$$