geno3141

Formula for the area of a circle:  Area  =  pi·diameter²/4

Old circumference = d   --->   old area  =  pi·d²/4

New circumference = 1.7d   --->   new area  =  pi·(1.7·d)²/4  =  2.89·pi·d²/4

To find the percentage increase, we first need to find the amount of increase:

                    amount of increase  =  new area - old area

                                                     =  2.89·pi·d²/4 - pi·d²/4

                                                     =  1.89·pi·d²/4

Now, we need to divide the amount of increase by the old area:

                  percentage increase  =  ( 1.89·pi·d²/4 ) / ( pi·d²/4 )

                                                     =  1.89

As a percentage:  189%  increase

AB = 7   --->   c = 7

AC = 2   --->   b = 2

BC = 8   --->   a = 8

Law of Cosines:  a²  =  b² + c² - 2·b·c·cos(A)

                            8²  =  2² + 7² - 2·2·7·cos(A)

                            64  =  4 + 49 - 28·cos(A)

                            64  =  53 - 28·cos(A)

                            11  =  - 28·cos(A)

                      -11/28  =  cos(A)

Let's multiply both the numerator and the denominator by 5^1/16 - 1:

The numerator becomes:  4·(5^1/16 - 1)

The denominator becomes:  (5^1/2 + 1)(5^1/4 + 1)(5^1/8 + 1)·[ (5^1/16 + 1)(5^1/16 -  1) ]

                                    =        (5^1/2 + 1)(5^1/4 + 1)(5^1/8 + 1)·[ (5^1/8 -  1) ]

re-group:                     =        (5^1/2 + 1)(5^1/4 + 1)·[ (5^1/8 + 1)(5^1/8 -  1) ]

                                    =        (5^1/2 + 1)(5^1/4 + 1)·[ (5^1/4 - 1) ]

re-group:                     =         (5^1/2 + 1)·[ (5^1/4 + 1)(5^1/4 - 1) ]

                                   =          (5^1/2 + 1)·[ (5^1/2 - 1) ]

                                    =         5 - 1

                                    =         4

Fraction:  4·(5^1/16 - 1) / 4  =  5^1/16 - 1

[ ( 5^1/16 - 1 ) + 1 ]⁴⁸  =  [ 5^1/16 ]⁴⁸  =  5³  =  125

a₀  =  1

a₁  =  -1

a₂  =  a₁ + 2·a₀  =  -1 + 2·1  =  -1 + 2  =  1

a₃  =  a₂ + 2·a₁  =  1 + 2·-1  =  1 - 2  =  -1

a₄  =  a3 + 2·a₂  =  -1 + 2·1  =  -1 + 2  =  1

a₅  =  a₄ + 2·a3  =  1 + 2·-1  =  1 - 2  =  -1

There seems to be a pattern ...

sec(x)  =  1/cos(x)          csc(x)  =  1/sin(x)

           sec(x)  =  -½·csc(x)

    [ 1/cos(x) ]  =  -½ · [ 1/sin(x) ]                       ( make the trig substitutions )

sin(x) / cos(x)  =  -½                                         ( multiply both sides by sin(x) )

            tan(x)  =   -½                                        ( tan(x)  =  sin(x) / cos(x) )

                   x  =  tan^-1(  -½ )

Can you finish?

                            sin(x) - cos(x)  =  0

                      [ sin(x) - cos(x) ]²  =  0²                    ( square both sides  

sin²(x) - 2sin(x)cos(x) + cos²(x)  =  0 

sin²(x) + cos²(x)  - 2sin(x)cos(x)  =  0                    ( rearrange )

                         1 - 2sin(x)cos(x)  =  0                    ( sin²(x) + cos²(x)  =  1 

                            - 2sin(x)cos(x)  =  -1

                              2sin(x)cos(x)  =  1

                                        sin(2x)  =  1                    ( sin(2x)   =  2sin(x)cos(x) )

                                               2x  =  sin^-1(1)

                                               2x  =  pi/2

                                                 x  =  pi/4                 ( principal root; extend to get the other roots )

The product of  a·cis(A)  and  b·cis(B)  is  (a·b)·cis(A + B).

For your problem, find the values of a, A, b, and B.

Then place them into the formula (and simplify).

Take out a sheet of graph paper.

1)  Plot the point (1,0); call this point A.

2)  Plot the point (2,2); call this point B.

3)  Plot the point (4,5); call this point C.

4)  Draw the line segments AB and BC; these will be the bottom and the right side of the parallelogram.

5)  Use this diagram to find point D.

6)  Find the length BC using the distance formula:  distance  =  sqrt[  (x₂ - x₁)² + (y₂ - y₁)² ] 

7)  Find the distance between BC and AD.

8)  Use the results of step 6 and step 7 to find the area of the parallelogram.

The first equation:  y  =  |x + 2| - 2  has its vertex at (-2,-2) and opens upward. The equation of its line of symmetry

is y  =  -2.

The second equation:  x  =  |y + 4|  has its vertex at (0,-4) and opens to the right. Can you find its line of symmetry?

If the original length is x and the original width is y, then the area will be:  A  =  xy.

If you increase the length by 100%, you are doubling the length to 2x.

If you increase the width by 100%, you are doubling the width to 2y.

The new area will be:  A = 2x·2y = 4xy.

The increase in area will be:  4xy - xy  =  3xy.

The percentage increase will be:  3xy / xy = 3, or 300%.