Chapter 2.6, Problem 13AYU

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

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

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

To determine

To find:

a. To express the circumference $C$ of circle as a function $x$, where $x$ is the length of a wire that is bend into a circle.

Answer to Problem 13AYU

Solution:

a. $C\left(x\right)=x$

Explanation of Solution

Given:

A wire of length $‘x’$ is bent into a circle.

Calculation:

a. To express the circumference $C$ of circle as a function $x$ :

Since the entire length ‘$x$ ’ is bent into a circle, the circumference, $C\left(x\right)=x$ .

To determine

To find:

b. To express the Area $A$ of the circle as a function of $x$.

Answer to Problem 13AYU

Solution:

b. $A=\frac{x^2}{4\pi }$

Explanation of Solution

Given:

A wire of length $‘x’$ is bent into a circle.

Calculation:

b. To express the $A$ of circle as a function $x$:

The circumference of the circle$=2\pi r=x$

$r=\frac{x}{2\pi }$

Area of the circle $A= \pi r^2=\pi {\left(\frac{x}{2\pi }\right)}^2$.

Therefore, $A\left(x\right)=\frac{x^2}{4\pi }$.