geno3141

And, what's the question?

When Suelyn counts from 1-2-3-4-5-6-7-8-9-8-7-6-5-4-3-2     (16 numbers)

                                           1-2-3-4-5-6-7-8-9-8-7-6-5-4-3-2     (anothe 16 numbers)

                                           etc.

In cycles of 16.

Divide 1000 by 16 and you get 62 with a remainder of 8.

So, there are 16 cycles, plus a remaining 8 numbers.

The 1000^th number will be the eighth number on the list  =  8.

Plan:

1) Using the area of the regular hexagon, find the length of one of its sides.

2) Find the number of degrees in each vertex angle of the regular hexagon.

3) Using steps 1) and 2), find the area of one of the white triangles.

4) Subtract the area of both of the white triangles from the area of the regular hexagon to get the area

        of the purple region.

1)  Formula for the area of a regular hexagon (A = area; a = length of a side):

          A  =  [ 3·sqrt(3)/2 ] ·a²

         90  =  [ 3·sqrt(3)/2 ] ·a²

       180  =  [ 3·sqrt(3) ] ·a²

         60  =  [ sqrt(3) ] ·a²

        a²  =  60 / sqrt(3)

        a²  =  60·sqrt(3)/3

        a²  =  20·sqrt(3)

         a  =  sqrt[ 20·sqrt(3) ]           <---  length of each side of the hexagon

2)  Formula for the number of degrees in each vertex angle of a regular polygon  (n = number of sides):

         degrees  =  (n - 2)·180^o / n

         degrees  =  (6 - 2)·180^o / 6

         degrees  =  120^o      <---  number of degrees in each vertex angle

3)  Formula to find the area of a white triangle 

        Area  =  ½·a·b·sin(C)

        Area  =  ½ · sqrt[ 20·sqrt(3) ]  ·sqrt[ 20·sqrt(3) ] · sin(120^o)

        Area  =  ½ · 20·sqrt(3) · sqrt(3)/2

        Area  =  15

        Area of both white triangles  =  30

4)  Area of the purple region:  area(hexagon) - area(both triangle)  =  90 - 30  =  60

The degree of a polynomial is the degree of its highest degree term.

Therefore, the degree of  f(x)  =  -7x⁴ + 3x³ + x - 5  is  4.

For the degree of  f(x) + g(x)  to be 1,  g(x) must have cancelled out the  -7x⁴  and the  3x³  terms but left the  x  term.

So, g(x) must include a  +7x⁴  term and a  -3x³  term but no other terms larger than x (and, not just a -x term, that would

cancel out all the x-terms and leave a degree of 0).

--->   the degree of  g(x)  must be 4.

Create a table with three columns

-- one column is the number of boxes that contain 1 piece of chocolate

-- one column is the number of boxes that contain 2 pieces of chocolate

-- one column is the number of boxes that contain 4 pieces of chocolate.

1 piece          2 pieces         4 pieces

______         _______         _______

   15                   0                     0

   13                   1                     0

   11                    2                    0

   11                    0                    1

    9                    3                     0

    9                    1                     1

    7                    4                     0

    7                    2                     1

    7                    0                     2

    5                     5                    0

    5                     3                    1

    5                     1                    2

etc.

Find the set of all possible values of  x + y  that fulfill  x·y  =  9.

When  x·y  =  9  . --->   y  =  9 / x      and     x + y  =  x + 9 / x   --->    x + y  =  (x² + 9) / x

The function:  f(x)  =  (x²​ + 9) / x

--  has a minimum in the first quadrant at 6, so its positive values are in  [6, infinity)

--  has a maximum in the thrid quadrant at -6, so its negative values are in (- infinity, -6]

One side of the triangle must be 6.

Then, if the second side is also 6, the third side can be either 6, 5, 4, 3, 2, or 1  --  6 possibilities.

If the second side is 5, the third side can be either 5, 4, 3, or 2  --  4 possibilities.

If the second side is 4, the third side can be either 4 or 3  --  2 possibilities.

The second side cannot be shorted than 4.

Did I miss any?

H(t)  =  -16·t² + 162

Find the value of t that makes H(t) = 0 ...

-16·t² + 162  =  0

          -16·t²  =  -162

                t²  =  -162/ -16

                t²  =  162/ 16

                 t  =  sqrt(162/16)     (ignore the negative value)

Let  a  be the length across the top of the red rectangle.

Let  b  be the length across the top of the blue rectangle.

Let  c  be the height across the left side of the red rectangle.

Let  d  be the height across the left side of the yellow rectangle.

Therefore, the area of the

   red rectangle is:      a·c  =  12

   blue rectangle is:     b·c  =  15

   yellow rectangle is:  a·d  =  8

   green rectangle is:  b·d  =  ??

a·c  =  12   --->   a  =  12/c

a·d  =  8     --->   a  =  8/d

Combining these two equations:  12/c  =  8/d   --->   12d  =  8c   --->   3d  =  2c   --->   c  =  3d/2

  b·c  =  15   --->   c  =  15/b

Combining  c  =  15/b  with  c  =  3d/2   --->   15/b  =  3d/2   --->   3bd  =  30   --->   b·d  =  10

Sketch a regular pentagon and one of its diagonals

--  that diagonal, along with two sides forms an isosceles triangle

You know two of the sides of that triangle; they are each 1

You can find the size of the vertex angle of the pentagon by using this formula:  (n - 2)·180^o / n

--  since  n = 5:  each vertex angle  =  (5 - 2)·180^o / 5  =  108^o.

To find the length of the diagonal (the base of the isosceles triangle), use the Law of Cosines:

--  c²  =  a² + b² - 2·a·b·cos(C)

--  c²  =  1² + 1² - 2·1·1·cos(108^o)

The value of c will be the length of a diagonal of a regular pentagon of unit side length ...