geno3141

My understanding of the problem is that 100 of these patches are used to create the quilt, 10 rows of 10 patches per row.

Since the area of the quilt is 1.44 m sq, the quilt must have dimensions of 1.2 m by 1.2 m. 

Since each row contains 10 patches and each row is 1.2 m long, each patch must have a width of 0.12 m or 12 cm.

By looking at the patch shown, the each black segment must be twice as wide as a yellow segment, so each yellow segment must be 3 by 3 cm, each black segment must be 3 by 6 cm and each white segment must be 6 by 6 cm.

To find how much of each color is needed, notice that the area of each patch is divided into four 3 x 3 yellow areas (= 36 cm sq), four 3 x 6 black areas ( = 72 cm sq) and one 6 x 6 white area ( = 36 cm sq). This means that 1/4 of each patch is yellow, 1/4 of each patch is white, and 1/2 of each patch is black.

Thus, 1/4 of the total quilt is yellow (1/4 of 1.44 m sq is 0.36 m sq) as is also the white area (another 0.36 m sq) but the black area is twice as much (o.72 m sq).

What is the smallest number that leaves a remainder of 5 when divided by 12, 14, and 18?

First, find the smallest number that is divisible by 12, 14, and 18:

   ---  factor each of the numbers:

                   12  =  2 x 2 x 3

                   14  =  2 x 7

                   18  =  2 x 3 x 3

  ---  the least common multiple of these numbers is  2 x 2 x 3 x 3 x 7  =  252

Thus, 252 is divisible by 12, 14, and 18.

If you want a remainder of 5, add 5 to that number:  252 + 5  =  257.

Factor:    4x² - 100

First, factor out the common term, 4:

               4(x² - 25)

               4(x + 5)(x - 5)

Find the domain of  y  =  sqrt( 16x² - 4 ).

The domain consists of all the values of x that will make the function defined. In this case, all the values of x that result in the expression  16x² - 4  >=  0.

To solve:  16x² - 4  >=  0

Divide all terms by 4:  4x² - 1  >=  0

--->                                   4x²  >=  1

--->                                     x²  >=  1/4

Find the square root of both sides and divide into two inequalities:

--->                     Either  x <= -1/2    or   x >= 1/2

Answer:      ( -infinity, -1/2 )  union  ( 1/2, infinity )

Verify the identity:  sqrt( [1 + sin(x)] / [1 - sin(x)] )  =  [1 + sin(x)] / | cos(x) |

Start with the left-hand side:

     sqrt( [1 + sin(x)] / [1 - sin(x)] )  

Multiply both the numerator and the denominator of the interior of this square root by the conjugate of the denominator

                                 the conjugate of  1 - sin(x)  is  1 + sin(x)

=  sqrt( [ ( 1 + sin(x) )( 1 + sin(x) ) ] / [ ( 1 - sin(x) ) / ( 1 + sin(x) ) ]

=  sqrt( [ ( 1 + sin(x) )² ] / [ 1 - sin²(x) ] )

=  sqrt( [ ( 1 + sin(x) )² ] / [ cos²(x) ] )

Divide this into two parts, the numerator and the denominator:

=    sqrt( ( 1 + sin(x) )² ) / sqrt ( cos²(x) )

Simplify the square roots:

=  ( 1 + sin(x) ) / | cos(x) |

f(x) and g(x) will be inverse functions when  f( g(x) )  =  x  and  g( f(x) )  =  x.

f(x)  =  sqrt( x - 4 )  [with x >= 4]

g(x)  =  x² + 4  [with x >= o]

f( g(x) )  =  f( x² + 4 )  =  sqrt( (x² + 4) - 4 )  =  sqrt( x² + 4 - 4 )  =  sqrt( x² ) )  =  x    [Since x >= 0]

g( f(x) )  =  g( sqrt( x - 4 ) )  =  sqrt( x - 4 )² + 4  =  (x - 4) + 4  =  x - 4 + 4  =  x

Since  f( g(x) )  =  g( f(x) )  =  0,  f(x) and g(x) are inverse functions.

Simplify:  (6x²y + 4xy² - 2y³ + 6) + (x²y + 3y³ - 5)

Rearrange, combining like terms:  (6x²y + x²y) + (4xy²) + (-2y³ + 3y³) + (6 - 5)

                                                           (7x²y) + (4xy²) + (1y³) + (1) 

                                                                 7x²y + 4xy² + y³ + 1

You have:  x³, 4x, 8x, and 8

What do you want to do with them?

The total number of refrigerators produced is the number of refrigerators produced each day times the number of days that of operation.

If 'r' represents the total number of refrigerators produced

and 'd' the number of days of production

and there are 51 refrigerators produced each day,

the appropriate formula is:  r  =  51·d

The probability when z <= -1.75 is approximately 0.401.

The probability when z <= 1.75 is approximately 0.9599.