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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 .

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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

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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

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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.

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Practice set c

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

14 and 15, amd –14 and –15

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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

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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 .

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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.

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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 .

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Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Uday
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
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?
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not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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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
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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