Chapter 2, Problem 73RE

(a)

To determine

To express: the area $A$ of the printed part of the page as a function of the width $x$ of the border.

Answer to Problem 73RE

The function is $A=4x^2-39x+93.5$ which represent the model of equation of area $A$ of printed part of the page as a width $x$ of the border of the page.

Explanation of Solution

Given information:

A page with dimensions of $8\frac{1}{2}$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

  

Calculation:

As by given figure, it’s obvious that the width of the inside printed rectangular portion can be obtained by subtraction twice from the original width of whole page as $x$ units margin is there in both left and right side of the printed area.

So the width of the printed area = $8.5-2x$ inches.

Similarly the length of the printed area = $11-2x$ inches

Now as,

Are of printed rectangular = length $x$ width

  $\left(8.5-2x\right)\times \left(11-2x\right)$

Now, on using FOIL rule,

  $\begin{array}{c}\left(8.5-2x\right)\times \left(11-2x\right)=8.5\times 11-8.5\times 2x-2x\times 11-2x\times \left(-2x\right)\\ =93.5-22x-17x+4x^2\\ =4x^2-39x+93.5\end{array}$

So, the function is $A=4x^2-39x+93.5$ which represent the model of equation of area $A$ of printed part of the page as a width $x$ of the border of the page.

(b)

To determine

To find: the domain and the range of $A$ .

Answer to Problem 73RE

The Domain of the given function = $\left[0,4.25\right)$

The range of printed page = $0<A<93.5$

Explanation of Solution

Given information:

A page with dimensions of $8\frac{1}{2}$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

  

Calculation:

As area is always positive so $A>0$ , or

  $4x^2-39x+9.5>0$

Now on using quadratic formula,

  $a=4,b=-39andc=+93.5$

  $\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{\left(-39\right)\pm \sqrt{{\left(-39\right)}^2-4\left(4\right)\left(+93.5\right)}}{2\left(4\right)}\\ =\frac{39\pm \sqrt{1521-1496}}{8}\\ =\frac{+39\pm \sqrt{25}}{8}\\ =\frac{39\pm 5}{8}\end{array}$

Now on taking + sign,

  $\begin{array}{c}x=\frac{39+5}{8}\\ =\frac{44}{8}\\ =5.5\end{array}$

Now on taking (-) sign,

  $\begin{array}{c}x=\frac{39-5}{8}\\ =\frac{34}{8}\\ =4.25\end{array}$

So, the Domain of the given function = $\left[0,4.25\right)$

It’s clear that the minimum area of printed page would be zero and maximum area would be equal to the dimension of its full length and width that is,

  $8.5\times 11=93.5$

So, the range of printed page = $0<A<93.5$

(c)

To determine

To find: the area of the printed page for borders of widths 1 inch, 1.2 inches, and 1.5 inches.

Answer to Problem 73RE

  $A=58.5\text{square}\text{inches},52.46\text{square}\text{inches},44\text{square}\text{inches}.$

Explanation of Solution

Given information:

A page with dimensions of $8\frac{1}{2}$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

  

Calculation:

Now at width $x=1$

  $\begin{array}{c}A=4{\left(1\right)}^2-39\times 1+93.5\\ =97.5-39\\ =58.5\text{square}\text{inches}\end{array}$

So, $A=58.5\text{square}\text{inches}$ .

When width $x=1.2$ inches

  $\begin{array}{c}A=4{\left(1.2\right)}^2-39\times 1.2+93.5\\ =5.75+46.7\\ =52.46\text{square}\text{inches}\end{array}$

So, $A=52.46\text{square}\text{inches}$

When width $x=1.5$ inches

  $\begin{array}{c}A=4{\left(1.5\right)}^2-39\times 1.5+93.5\\ =9+35\\ =44\text{square}\text{inches}\end{array}$

So, $A=44\text{sqaure}\text{inches}$

(d)

To determine

To sketch: the graph of the function $A=A\left(x\right)$ .

Explanation of Solution

Given information:

A page with dimensions of $8\frac{1}{2}$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

  

Calculation:

The graph of the quadratic equation, which is a parabola, showing area i.e

  $A=4x^2-39x+93.5$ is shown as under when sketched using a graphics utility program: