3.3 Quadratic equations: applications  (Page 2/4)

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

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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