Chapter 3.1, Problem 74E

To determine

To find: The function that models the total area enclosed by two squares and also the value of $x$ that minimizes the total area of two squares.

Answer to Problem 74E

The function that models the total area enclosed by two squares is equal to $\text{A}={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ and also the value of $x$ that minimizes the total area of two squares is $x=5$ .

Explanation of Solution

Given information:

A wire $10\text{cm}$ long is cut into two pieces, one of length $x$ and other of length $10-x$ , as shown in figure below. Each piece is bent in shape of a square.

  

Calculation:

As given the length of one wire is $x$ and the wire is bent in shape of square. So, the length of each side of square is $\frac{x}{4}$ .

As given the length of other wire is $10-x$ and the wire is bent in shape of square. So, the length of each side of second square is $\frac{10-x}{4}$ .

The area of the first square is equal to ${\left(\frac{x}{4}\right)}^2$ and the area of the second square is equal to ${\left(\frac{10-x}{4}\right)}^2$ . The total area of the squares is equal to $\text{A}={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ .

So, the function of the total area of the squares is given by $g\left(x\right)={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ . To minimize the total area of squares graph the function $g\left(x\right)={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ .

To graph a function $y={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{÷}\boxed{4}\boxed{)}\boxed{^}\boxed{2}\boxed{+}\boxed{(}\boxed{(}\boxed{10}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{4}\boxed{)}\boxed{^}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}/4\right)^2+\left(\left(10-\text{X}\right)/4\right)^2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the area function has minimum value at $x=5$ .

Therefore, the function that models the total area enclosed by two squares is equal to $\text{A}={\left(\frac{x}{4}\right)}^2+{\left(\frac{10-x}{4}\right)}^2$ and also the value of $x$ that minimizes the total area of two squares is $x=5$ .