9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

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.

(a)

Expert Solution

Answer to Problem 73RE

The function is $A=4x2−39x+93.5$ 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 $812$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

Precalculus, Chapter 2, Problem 73RE , additional homework tip 1

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

$(8.5−2x)×(11−2x)$

Now, on using FOIL rule,

$(8.5−2x)×(11−2x)=8.5×11−8.5×2x−2x×11−2x×(−2x)=93.5−22x−17x+4x2=4x2−39x+93.5$

So, the function is $A=4x2−39x+93.5$ 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 $A$ .

(b)

Expert Solution

Answer to Problem 73RE

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

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

Explanation of Solution

Given information:

A page with dimensions of $812$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

Precalculus, Chapter 2, Problem 73RE , additional homework tip 2

Calculation:

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

$4x2−39x+9.5>0$

Now on using quadratic formula,

$a=4,b=−39 and c=+93.5$

$x=−b±b2−4ac2a=(−39)±(−39)2−4(4)(+93.5)2(4)=39±1521−14968=+39±258=39±58$

Now on taking + sign,

$x=39+58=448=5.5$

Now on taking (-) sign,

$x=39−58=348=4.25$

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

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,

$8.5×11=93.5$

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.

(c)

Expert Solution

Answer to Problem 73RE

$A=58.5 square inches, 52.46 square inches, 44 square inches.$

Explanation of Solution

Given information:

A page with dimensions of $812$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

Precalculus, Chapter 2, Problem 73RE , additional homework tip 3

Calculation:

Now at width $x=1$

$A=4(1)2−39×1+93.5=97.5−39=58.5 square inches$

So, $A=58.5 square inches$ .

When width $x=1.2$ inches

$A=4(1.2)2−39×1.2+93.5=5.75+46.7=52.46 square inches$

So, $A=52.46 square inches$

When width $x=1.5$ inches

$A=4(1.5)2−39×1.5+93.5=9+35=44 square inches$

So, $A=44 sqaure inches$

(d)

To determine

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

(d)

Expert Solution

Explanation of Solution

Given information:

A page with dimensions of $812$ inches by 11 inches has a border of uniform width $x$ surrounding the printed matter of the page, as shown in the figure.

Precalculus, Chapter 2, Problem 73RE , additional homework tip 4

Calculation:

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

$A=4x2−39x+93.5$ is shown as under when sketched using a graphics utility program:

Precalculus, Chapter 2, Problem 73RE , additional homework tip 5

Chapter 2, Problem 72RE

Chapter 2, Problem 74RE

Chapter 2 Solutions

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

Concept explainers

Videos

To express: the area A<m:math><m:mi>A</m:mi></m:math> of the printed part of the page as a function of the width x<m:math><m:mi>x</m:mi></m:math> of the border.

Answer to Problem 73RE

Explanation of Solution

To find: the domain and the range of A<m:math><m:mi>A</m:mi></m:math> .

Answer to Problem 73RE

Explanation of Solution

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

Explanation of Solution

To sketch: the graph of the function A=A(x)<m:math><m:mrow><m:mi>A</m:mi><m:mo>=</m:mo><m:mi>A</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> .

Explanation of Solution

Chapter 2 Solutions

Additional Math Textbook Solutions

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

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

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