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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, 2 x + 4 = the length.     A rectangle with length labeled two x plus four and width labeled x.

Step  2 : The area of a rectangle is defined to be the length of the  rectangle times the width of the rectangle . Thus, x ( 2 x + 4 ) = 30
Step  3 : x ( 2 x + 4 ) = 30 2 x 2 + 4 x = 30 2 x 2 + 4 x 30 = 0 Divide each side by 2 . x 2 + 2 x 15 = 0 Factor . ( x + 5 ) ( x 3 ) = 0 x = 5 , 3 x = 5 has  no physical meaning so we disregard it . Check  x = 3. x = 3 2 x + 4 = 2 · 3 + 4 = 10
Step  4 : x ( 2 x + 4 ) = 30 Is this correct? 3 ( 2 · 3 + 4 ) = 30 Is this correct? 3 ( 6 + 4 ) = 30 Is this correct? 3 ( 10 ) = 30 Is this correct? 30 = 30 Yes, this is correct .
Step  5 : Width = 3 inches and length = 1 0  inches .

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 = 1 2 b · h .
A triangle with a line that is passing through one of its vertices and is perpendicular to the side opposite to this vertex. This line is labeled as 'height'. The side opposite to the vertex is labeled as base.

height = 6 meters, base = 8 meters

Sample set c

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

Step 1 : Let  x = the smaller integer . x + 1 = the next integer . Step 2 : x ( x + 1 ) = 156 Step 3 : x ( x + 1 ) = 156 x 2 + x = 156 x 2 + x 156 = 0 ( x 12 ) ( x 13 ) = 0 x = 12 , 13
This factorization may be hard to guess. We could also use the quadratic formula.

x 2 + x 156 = 0 a = 1 , b = 1 , c = 156 x = 1 ± 1 2 4 ( 1 ) ( 156 ) 2 ( 1 ) = 1 ± 1 + 624 2 = 1 ± 25 2 1 ± 25 2 = 24 2 = 12 and 1 25 2 = 26 2 = 13 x = 12 , 13 Check 12, 13 and  13, 12 . x + 1 = 13 , 12 Step 4 : If x = 12 : 12 ( 2 + 1 ) = 156 Is this correct? 12 ( 13 ) = 156 Is this correct? 156 = 156 Is this correct? If x = 13 13 ( 13 + 1 ) = 156 Is this correct? 13 ( 12 ) = 156 Is this correct? 156 = 156 Yes, this is correct . Step 5 : There are t w o solutions: 12 , 13 and 13 , 12.

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?

A square with smaller squares drawn in each corner of the larger square. The smaller squares are labeled with a length of two and the length between the two smaller squares is labeled as x minus four. The length of the large square is labeled as x.

Step 1: Let  x = the length  ( and width )  of the piece of cardboard .
Step 2: The volume of a rectangular box is V = ( length ) ( width ) ( height ) 8 = ( x 4 ) ( x 4 ) 2
Step  3 : 8 = ( x 4 ) ( x 4 ) 2 8 = ( x 2 8 x + 16 ) 2 8 = 2 x 2 16 x + 32 2 x 2 16 x + 24 = 0 Divide each side by 2 . x 2 8 x + 12 = 0 Factor . ( x 6 ) ( x 2 ) = 0 x = 6 , 2
      x cannot equal 2 (the cut would go through the piece of cardboard). Check x = 6.
Step  4 : ( 6 4 ) ( 6 4 ) 2 = 8 Is this correct? ( 2 ) ( 2 ) 2 = 8 Is this correct? 8 = 8 Yes, this is correct .
Step  5 : The piece of cardboard should be 6 inches by 6 inches .

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 + 0.1 t + 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 + 0.1 t + 4.2 specifies the level t years from now, we have t = 0.

   A = 0.3 t 2 + 0.1 t + 4.2
   A = 4.2

(b) How many years from now will the level of carbon monoxide be at 8 parts per million?
   Step  1 : t = the number of years when the level is 8 .
   Step  2 : 8 = 0.3 t 2 + 0.1 t + 4.2
   Step  3 : 8 = 0.3 t 2 + 0.1 t + 4.2 0 = 0.3 t 2 + 0.1 t 3.8 This does not readily factor, so  we'll use the quadratic formula . a = 0.3 , b = 0.1 , c = 3.8 t = 0.1 ± ( 0.1 ) 2 4 ( 0.3 ) ( 3.8 ) 2 ( 0.3 ) = 0.1 ± 0.01 + 4.56 0.6 = 0.1 ± 4.57 0.6 = 0.1 ± 2.14 0.6 t = 3.4 and 3.73
   t = 3.73 has no physical meaning .  Check t = 3.4 Step 4 : This value of  t  has been rounded to the nearest tenth .  It does check  ( pretty closely ) . Step 5 : About 3 . 4 years from now the carbon monoxide level will be 8 .

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
learn
Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
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