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

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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