CPhill

The total  number of integers   =  250 - 25 + 1   = 226

And the perfect cubes  in this group are    27, 64, 125, 216   = 4

So...the number of integers that are not perfect cubes  =  [ 226 - 4 ] =  222

On each toss  we have 8 possibilities

So...the number of possible sequences possible  = 8^6  = 262144 

x^2 +3x - 28  factors as  ( x +7) ( x - 4)

x^2  - 7x  - 12  factors as  ( x - 3) ( x - 4)

So....the (x - 4)  factors will "cancel" in the numerator /denomnator  and we  are left with

[ x + 7 ]

_______

 [ x - 3 ]

The radius of the Earth  = 8000/2  = 4000 miles

We  can  figure the distance  to the horizon  using a right triangle

The distance from  the center of the Earth to any object above the Earth's surface  = the hypotenuse = 4000  + h      where h is the height of the object [in  miles ] above the Earth's surface

And the radius of the Earth  is equal to one leg of the right triangle

And the distance from the object to the horizon, D,  is the other leg

So...we have this general formula to determine D

D  = √ [ (4000 + h)^2  - 4000^2]

So....

(a)  D  = √ [ (4000 + 1)^2  - 4000^2 ]  ≈ 89.4 miles

(b)  D  =   √ [ (4000 + 6)^2  - 4000^2 ]  ≈  219.2  miles

(c)  D  = √ [ (4000 + 100)^2  - 4000^2 ]  = 900 miles

(a)  Call x the additional consecutive number of FTs she needs to make

We have this equation

[12 + x ] / [ 15 +  x ] = .85

(b)   Multiply both sides by  15 + x

12 + x  = .85 [ 15 + x]

12 + x  =  12.75  + .85x      subtract  .85x, 12 from both sides

.15x  = .75     divide both sides by .15

x  = 5  additional consecutive FTs

Let  the total number of skateboards = X

Let the total number of scooters  = Y

And we know that

(1/4)X  = (2/3)Y       multiply both sides by 3/2

(3/8)X  =  Y

So fraction  of  red skateboards and scooters to the total  =

[ 1/4X + (2/3)Y ]  /  [  X + Y  ]  =

[ 1/4X  + 1/4X  ]  /  [ X + (3/8)X ] =

[ (1/2) X ]  / [ (8/8)X + (3/8)X ] =

[   (1/2)X ]  / [ (11/8) X ] =  (cancel the  X's )

(1/2)  / (11/8)  =  (invert the second fraction and multiply )

(1/2) * (8 /11)  =

8 / 22  =

4 /11

The sides of the square must be √6 cm  each

Let x be the distance between the legs of the black triangle and the side of the square

The white area is composed of a square of sides of  x  and two rectangles with dimensions of x and [ √6 - x ]   

So.....the white area  =    x^2 + 2x ( √6 - x )

And the area of the  black triangle  =  ( √6 - x )^2 / 2  = [ 6 - [2√6]x  + x^2] / 2

Since these areas are equal  we have that

x^2 + 2x ( √6 - x )  = [ 6  - [2√6]x + x^2] / 2

2 [ x^2  + 2x (√6 - x) ] = 6 - [2√6]x + x^2 

2x^2 + 4x(√6 - x ]   = 6 - [2√6]x  + x^2

x^2 + [4√6] x - 4x^2  = 6 - [2√6]x

-3x^2 + [ 6√6]x   = 6

3x^2  - (6√6]x  = -6

x^2 - [2√6]x = -2     complete the square

x^2  - [2√6]x  +  6  =  -2 + 6

( x - √6)^2  = 4    take both roots

x - √6  = 2        or     x - √6  = -2

x = √6 + 2       or     x  =  √6  - 2

(reject)                           (accept)

So  ...let the original position of A   = (0, 0)

And the new position  of A  =  ( √6  - ( √6  - 2) ,  √6 - ( √6 - 2) )  = ( 2 , 2)

And its distance from its original position is

√ [ 2^2  +  2^2  ]  =

√8  =

2√2

Correct !!!!

Let's simplify the denominator, first

1/6  +  1 / [x + 1]  =

[ x + 1  + 6 ]  / [ 6(x + 1)] =

[ x + 7]  / [ 6(x + 1) ]

So we have

    5  / 1

_________________       [flip the fraction in the denominator and multiply ]

  [x + 7 ] / [ 6 (x + 1)

5/1  *  [ 6 ( x + 1) ] /  ( x + 7 ]  =

30(x + 1)              30x + 30

_______  =         ________

  x + 7                    x +  7

a)What is the area between the graph of y=6f(x-6) and the x-axis?

The (x - 6) part just shifts the graph to the right by 6 units.....this does not change the area

The  "6"  part  makes every point on the graph 6 times as high..so the area  is

6 * 10   = 60 units^2

b)What is the area between the graph of y=1/2f(-x) and the x-axis?

The (-x)  part rotates the graph about the y axis....the area is unchanged

The (1/2) part makes every point on the graph 1/2 as high....so the area is

(1/2) * 10   =  5 units^2

c)What is the area between the graph of y=f(x+2) and the x-axis?

This just shifts the graph 2 uits to the left....the area is unchanged