CPhill

a) No condition is imposed

We can use any of the seven letters in each position...so

7 * 7 * 7 * 7  =  2401 "words"

b) No letter can be repeated in a word.

We have 7 ways to pick the first letter, 6 ways to pick the second, 5 ways to pick the third and 4 ways to pick the fourth............so   7 * 6 * 5 * 4  =  840 "words"

c) Each word must begin with the letter A

If we can repeat letters, the other three positions can be filled in 7 * 7 * 7 =  343 ways....so..we have 343 "words"

If we cannot repeat letters we have   6 * 5 * 4  ways to fill the other three positions = 120 "words"

d) The letter C must be at the end.

Same answers as  (c)

e) The second letter must be a vowel.

If we can repeat letters....we have 7 ways to fill three of the positions and 2 ways to fill the remaining one [ since we have only two vowels ]...so....   7 * 2 * 7 * 7  = 686 "words"

If we cannot repeat letters, we have 7 * 2 * 6 * 5  =  420 "words"

Since AOB is 90°, the  whole area is just the area of 1/4 of a circle = pi (8)^2 / 4  =

64 pi / 4  =  16 pi  units^2       (1)

The area of the triangular part of the figure is(1/2) * 8^2  =  64/2  = 32 units^2     (2)

So...the purple area is just   (1)  - (2)  =   [16 pi - 32 ] units^2 

The other leg  = 5

And the area of the base is    (1/2) (5) (12)  =  (1/2) * 60  =  30 units^2

And the volume of the prism is

Base area * Height  =    30  * 10   =  300 units cubed

Let the smaller circle's radius  = r  ⇒ circumference =  2pi r

And the larger circle has a radius twice the smaller  = 2r  ⇒  circumerence = 2pi (2r)  =

4 pi r

So...the ratio of their circumferences  =

[ 2 pi r ] : [ 4 pi r ]  =  2 : 4  =  1 : 2

ΔAXC   is similar to  ΔAYB  by  AA congruency

So

XC /AC = YB / AB

XC / 70  = 60 / 75 

XC / 70  =  4 / 5       

XC  =  (4 / 5)  * 70  =  56

Very nice,  AT  !!!

Area of square  = 1 unit^2

The circle has a radius of  (1/2)

So...its area is :    pi   (1/2)^2  =  pi / 4  units^2

So    area of circle to area of square  =   [ pi / 4] : 1  =   pi / 4  ≈ .785  ≈  79%

x^4 + 5x^2 + 6  = 0

We  can factor this as

(x^2  + 3) (x^2 + 2)  = 0

Setting each factor to 0  we have that

x^2  + 3  =  0               x^2  + 2  = 0

x^2  =  -3                     x^2  =  -2

Note that no real numbers make these true, so, there are 4 complex zeroes

When a graph just "kisses" the x axis, we will have 2n zeroes at that point

So...let's assume that  x = 2  we have two zeroes

Thus,  let us guess that   (x - 2)^2  =  x^2 - 4x + 4   divides the polynomial

                           3x^2   +  x     +  2   

  x^2 - 4x + 4   [  3x^4  -11x^3  +  10x^2  -  4x + 8  ]

                           3x^4  -12x^3  +  12x^2

                           ______________________________

                                      x^3    -      2x^2     -    4x   +  8

                                      x^3    -      4x^2     +   4x

                                     ________________________

                                                        2x^2   -     8x  + 8

                                                        2x^2   -     8x  + 8 

                                                        ________________

So...the factorization is   ( 3x^2   +  x  +  2  )  ( x - 2) ^2

Correct....!!!!