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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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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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