Q. 14 - Precalculus Enhanced with Graphing Utilities

Question

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

Part (a): Express the perimeter p of the square as a function of x.

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

Step-by-Step Solution

Answer

Part (a): Perimeter of the square as a function is $p\left(x\right)=x$.

Part (b): Area of the square as a function is $A\left(x\right)=\frac{x^2}{16}$.

Part (a) Step 1. Given information.

Consider the given question,

Length of the wire x is equal to the perimeter of the square which is $p\left(x\right)=x$.

Part (b) Step 1. Given information.

Consider the given question,

Length of the side of the square a,

$$\begin{gathered} a=\frac{p\left(x\right)}{4} \\ =\frac{x}{4} \end{gathered}$$

We know area of the square is $A=a^2$.

Substitute the values,

$$\begin{gathered} A\left(x\right)={\left(\frac{x}{4}\right)}^2 \\ A\left(x\right)=\frac{x^2}{16} \end{gathered}$$