# 10.6 Applications  (Page 2/4)

 Page 2 / 4

## Sample set b

The length of a rectangle is 4 inches more than twice its width. The area is 30 square inches. Find the dimensions (length and width).

Step 1:   Let $x=$ the width. Then, $2x+4=$ the length. $\begin{array}{lll}\text{Step\hspace{0.17em}}2:\hfill & \hfill & \text{The\hspace{0.17em}area\hspace{0.17em}of\hspace{0.17em}a\hspace{0.17em}rectangle\hspace{0.17em}is\hspace{0.17em}defined\hspace{0.17em}to\hspace{0.17em}be\hspace{0.17em}the\hspace{0.17em}length\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}rectangle\hspace{0.17em}times\hspace{0.17em}the\hspace{0.17em}width\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}rectangle}\text{.\hspace{0.17em}Thus,}\hfill \\ \hfill & \hfill & x\left(2x+4\right)=30\hfill \end{array}$
$\begin{array}{lllllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & \hfill x\left(2x+4\right)& =\hfill & 30\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 2{x}^{2}+4x& =\hfill & 30\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 2{x}^{2}+4x-30& =\hfill & 0\hfill & \hfill & \text{Divide\hspace{0.17em}each\hspace{0.17em}side\hspace{0.17em}by\hspace{0.17em}2}.\hfill \\ \hfill & \hfill & \hfill {x}^{2}+2x-15& =\hfill & 0\hfill & \hfill & \text{Factor}.\hfill \\ \hfill & \hfill & \hfill \left(x+5\right)\left(x-3\right)& =\hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill x& =\hfill & -5,\text{\hspace{0.17em}}3\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill x& =\hfill & -5\hfill & \hfill & \text{has\hspace{0.17em}\hspace{0.17em}no\hspace{0.17em}physical\hspace{0.17em}meaning\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}disregard\hspace{0.17em}it}\text{.\hspace{0.17em}Check\hspace{0.17em}}x=3.\hfill \\ \hfill & \hfill & \hfill x& =\hfill & 3\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 2x+4=2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3+4& =\hfill & 10\hfill & \hfill & \hfill \end{array}$
$\begin{array}{llllll}\text{Step\hspace{0.17em}}4:\hfill & \hfill & \hfill x\left(2x+4\right)& =\hfill & 30\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 3\left(2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3+4\right)& =\hfill & 30\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 3\left(6+4\right)& =\hfill & 30\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 3\left(10\right)& =\hfill & 30\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 30& =\hfill & 30\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}$
$\begin{array}{lll}\text{Step\hspace{0.17em}}5:\hfill & \hfill & \text{Width}=\text{3\hspace{0.17em}inches\hspace{0.17em}and\hspace{0.17em}length}=\text{1}0\text{\hspace{0.17em}inches}.\hfill \end{array}$

## Practice set b

The length of a rectangle is 3 feet more than twice its width. The area is 14 square feet. Find the dimensions.

width = 2 feet, length = 7 feet

The area of a triangle is 24 square meters. The base is 2 meters longer than the height. Find the base and height. The formula for the area of a triangle is $A=\frac{1}{2}b·h.$ height = 6 meters, base = 8 meters

## Sample set c

The product of two consecutive integers is 156. Find them.

$\begin{array}{l}\begin{array}{lllll}\text{Step\hspace{0.17em}1}:\hfill & \hfill & \hfill \text{Let\hspace{0.17em}}x& =\hfill & \text{the\hspace{0.17em}smaller\hspace{0.17em}integer}\text{.}\hfill \\ \hfill & \hfill & \hfill x+1& =\hfill & \text{the\hspace{0.17em}next\hspace{0.17em}integer}\text{.}\hfill \\ \text{Step\hspace{0.17em}2}:\hfill & \hfill & \hfill x\left(x+1\right)& =\hfill & 156\hfill \\ \text{Step\hspace{0.17em}3}:\hfill & \hfill & \hfill x\left(x+1\right)& =\hfill & 156\hfill \\ \hfill & \hfill & \hfill {x}^{2}+x& =\hfill & 156\hfill \\ \hfill & \hfill & \hfill {x}^{2}+x-156& =\hfill & 0\hfill \\ \hfill & \hfill & \hfill \left(x-12\right)\left(x-13\right)& =\hfill & 0\hfill \\ \hfill & \hfill & \hfill x& =\hfill & 12,\text{\hspace{0.17em}}-13\hfill \end{array}\\ \end{array}$
This factorization may be hard to guess. We could also use the quadratic formula.

$\begin{array}{l}{x}^{2}+x-156=0\\ \begin{array}{lllll}a=1,\hfill & \hfill & b=1,\hfill & \hfill & c=-156\hfill \end{array}\\ \begin{array}{lllll}\hfill x& =\hfill & \frac{-1±\sqrt{{1}^{2}-4\left(1\right)\left(-156\right)}}{2\left(1\right)}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{-1±\sqrt{1+624}}{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{-1±25}{2}\hfill & \hfill & \frac{-1±25}{2}=\frac{24}{2}=12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{-1-25}{2}=\frac{-26}{2}=-13\hfill \\ \hfill x& =\hfill & 12,\text{\hspace{0.17em}}-13\hfill & \hfill & \text{Check\hspace{0.17em}12,\hspace{0.17em}13\hspace{0.17em}and\hspace{0.17em}}-\text{13,\hspace{0.17em}12}\text{.}\hfill \\ \hfill x+1& =\hfill & 13,\text{\hspace{0.17em}}-12\hfill & \hfill & \hfill \end{array}\\ \end{array}$ $\begin{array}{l}\begin{array}{rrrrrrrr}\hfill \text{Step}\text{\hspace{0.17em}}4:& \hfill & \hfill \text{If}\text{\hspace{0.17em}}x=12:& \hfill & \hfill 12\left(2+1\right)& \hfill =& \hfill 156& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill & \hfill & \hfill 12\left(13\right)& \hfill =& \hfill 156& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill & \hfill & \hfill 156& \hfill =& \hfill 156& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill \text{If}\text{\hspace{0.17em}}x=-13& \hfill & \hfill -13\left(-13+1\right)& \hfill =& \hfill 156& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill & \hfill & \hfill -13\left(-12\right)& \hfill =& \hfill 156& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill & \hfill & \hfill 156& \hfill =& \hfill 156& \hfill \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\end{array}\hfill \\ \begin{array}{lllll}\mathrm{Step}\text{\hspace{0.17em}}5:\hfill & \hfill & \text{There}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}two\text{\hspace{0.17em}}\text{solutions:}\hfill & \hfill & 12,\text{\hspace{0.17em}}13\text{\hspace{0.17em}}\text{and}-13,\text{\hspace{0.17em}}-12.\hfill \end{array}\hfill \end{array}$

## Practice set c

The product of two consecutive integers is 210. Find them.

14 and 15, amd –14 and –15

Four is added to an integer and that sum is tripled. When this result is multiplied by the original integer, the product is −12. Find the integer.

–2

## Sample set d

A box with no top and a square base is to be made by cutting out 2-inch squares from each corner and folding up the sides of a piece of a square cardboard. The volume of the box is to be 8 cubic inches. What size should the piece of cardboard be? $\begin{array}{lll}\text{Step\hspace{0.17em}1:}\hfill & \hfill & \text{Let\hspace{0.17em}}x=\text{the\hspace{0.17em}length\hspace{0.17em}}\left(\text{and\hspace{0.17em}width}\right)\text{\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}piece\hspace{0.17em}of\hspace{0.17em}cardboard}.\hfill \end{array}$
$\begin{array}{lll}\text{Step\hspace{0.17em}2:}\hfill & \hfill & \text{The\hspace{0.17em}volume\hspace{0.17em}of\hspace{0.17em}a\hspace{0.17em}rectangular\hspace{0.17em}box\hspace{0.17em}is}\hfill \\ \hfill & \hfill & V=\left(\text{length}\right)\text{\hspace{0.17em}}\left(\text{width}\right)\text{\hspace{0.17em}}\left(\text{height}\right)\hfill \\ \hfill & \hfill & 8=\left(x-4\right)\left(x-4\right)2\hfill \end{array}$
$\begin{array}{lllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & 8=\left(x-4\right)\left(x-4\right)2\hfill & \hfill & \hfill \\ \hfill & \hfill & 8=\left({x}^{2}-8x+16\right)2\hfill & \hfill & \hfill \\ \hfill & \hfill & 8=2{x}^{2}-16x+32\hfill & \hfill & \hfill \\ \hfill & \hfill & 2{x}^{2}-16x+24=0\hfill & \hfill & \text{Divide\hspace{0.17em}each\hspace{0.17em}side\hspace{0.17em}by\hspace{0.17em}2}\text{.}\hfill \\ \hfill & \hfill & {x}^{2}-8x+12=0\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \hfill & \hfill & \left(x-6\right)\left(x-2\right)=0\hfill & \hfill & \hfill \\ \hfill & \hfill & x=6,\text{\hspace{0.17em}}2\hfill & \hfill & \hfill \end{array}$
$x$ cannot equal 2 (the cut would go through the piece of cardboard). Check $x=6.$
$\begin{array}{rrrrrr}\hfill \text{Step\hspace{0.17em}}4:& \hfill & \hfill \left(6-4\right)\left(6-4\right)2& \hfill =& \hfill 8& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill \left(2\right)\left(2\right)2& \hfill =& \hfill 8& \hfill \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\\ \hfill & \hfill & \hfill 8& \hfill =& \hfill 8& \hfill \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\end{array}$
$\begin{array}{rrr}\hfill \text{Step\hspace{0.17em}}5:& \hfill & \hfill \text{The\hspace{0.17em}piece\hspace{0.17em}of\hspace{0.17em}cardboard\hspace{0.17em}should\hspace{0.17em}be\hspace{0.17em}6\hspace{0.17em}inches\hspace{0.17em}by\hspace{0.17em}6\hspace{0.17em}inches}.\end{array}$

## Practice set d

A box with no top and a square base is to be made by cutting 3-inch squares from each corner and folding up the sides of a piece of cardboard. The volume of the box is to be 48 cubic inches. What size should the piece of cardboard be?

10 in. by 10 in.; 2 by 2 is not physically possible.

## Sample set e

A study of the air quality in a particular city by an environmental group suggests that $t$ years from now the level of carbon monoxide, in parts per million, in the air will be

$A=0.3{t}^{2}+0.1t+4.2$

(a) What is the level, in parts per million, of carbon monoxide in the air now?

Since the equation $A=0.3{t}^{2}+0.1t+4.2$ specifies the level $t$ years from now, we have $t=0.$

$A=0.3{t}^{2}+0.1t+4.2$
$A=4.2$

(b) How many years from now will the level of carbon monoxide be at 8 parts per million?
$\begin{array}{rrrrr}\hfill \text{Step\hspace{0.17em}}1:& \hfill & \hfill t& \hfill =& \hfill \text{the\hspace{0.17em}number\hspace{0.17em}of\hspace{0.17em}years\hspace{0.17em}when\hspace{0.17em}the\hspace{0.17em}level\hspace{0.17em}is\hspace{0.17em}8}.\end{array}$
$\begin{array}{rrrrr}\hfill \text{Step\hspace{0.17em}}2:& \hfill & \hfill 8& \hfill =& \hfill 0.3{t}^{2}+0.1t+4.2\end{array}$
$\begin{array}{lllllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & 8\hfill & =\hfill & 0.3{t}^{2}+0.1t+4.2\hfill & \hfill & \hfill \\ \hfill & \hfill & 0\hfill & =\hfill & 0.3{t}^{2}+0.1t-3.8\hfill & \begin{array}{l}\text{This\hspace{0.17em}does\hspace{0.17em}not\hspace{0.17em}readily\hspace{0.17em}factor,\hspace{0.17em}so\hspace{0.17em}}\\ \text{we'll\hspace{0.17em}use\hspace{0.17em}the\hspace{0.17em}quadratic\hspace{0.17em}formula}\text{.}\end{array}\hfill & \hfill \\ \hfill & \hfill & a\hfill & =\hfill & 0.3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=0.1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=-3.8\hfill & \hfill & \hfill \\ \hfill & \hfill & t\hfill & =\hfill & \frac{-0.1±\sqrt{{\left(0.1\right)}^{2}-4\left(0.3\right)\left(-3.8\right)}}{2\left(0.3\right)}\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & =\hfill & \frac{-0.1±\sqrt{0.01+4.56}}{0.6}=\frac{-0.1±\sqrt{4.57}}{0.6}\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & =\hfill & \frac{-0.1±2.14}{0.6}\hfill & \hfill & \hfill \\ \hfill & \hfill & t\hfill & =\hfill & 3.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-3.73\hfill & \hfill & \hfill \end{array}$
$\begin{array}{lll}\hfill & \hfill & t=-3.73\text{\hspace{0.17em}}\text{has\hspace{0.17em}no\hspace{0.17em}physical\hspace{0.17em}meaning}.\text{\hspace{0.17em}Check}\text{\hspace{0.17em}}t=3.4\hfill \\ \text{Step\hspace{0.17em}4}:\text{\hspace{0.17em}}\hfill & \hfill & \text{This\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}t\text{\hspace{0.17em}has\hspace{0.17em}been\hspace{0.17em}rounded\hspace{0.17em}to\hspace{0.17em}the\hspace{0.17em}nearest\hspace{0.17em}tenth}.\text{\hspace{0.17em}It\hspace{0.17em}does\hspace{0.17em}check\hspace{0.17em}}\left(\text{pretty\hspace{0.17em}closely}\right).\text{\hspace{0.17em}}\hfill \\ \text{Step\hspace{0.17em}5}:\text{\hspace{0.17em}}\hfill & \hfill & \text{About\hspace{0.17em}3}.\text{4\hspace{0.17em}years\hspace{0.17em}from\hspace{0.17em}now\hspace{0.17em}the\hspace{0.17em}carbon\hspace{0.17em}monoxide\hspace{0.17em}level\hspace{0.17em}will\hspace{0.17em}be\hspace{0.17em}8}.\text{\hspace{0.17em}}\hfill \end{array}$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications? By By By  By  By By By By  