CPhill

2x^2  + mx  + 8

If this has two distinctreal roots, then the discriminant  is  >  0

Sp

b^2  - 4ac  > 0

m^2  - 4(2)(8)  > 0

m^2 - 64  > 0

m^2  >  64

The possible values for m are

( - inf, -8)  U   (8, inf )

Let R1  be the radius of the first circle

And let R2 be the measure of the second circle

So

Arc length  =   radius * theta ( in rads )

So....since the arc lenghts are equal....

R1 * pi/2   = R2 * pi/3

R1 / 2  =  R2 / 3

R1  =  (2/3)R2

So...the area of  the first circle  is

[ pi * (R1) ^2 ]  =    pi *[  (2/3) R2 ] ^2  =  pi  (4/9) (R2)^2

And the area of the second circle  is

pi [ R2]^2  

So  the ratio  of the area  of the first cirlce to the second  is

[ pi * (4/9) (R2)^2 ]               4

_______________  =       ____

[ pi * (R2)^2 ]                        9

Note that the  "formula"  for determining the measure of an interior angle in any regular polygon  is

180 (n - 2) / n

Note that an integer value for an interior angle's measure will occur  whenever  n  divides 180  evenly

So regular polygons  with sides of 3,4,5,6 and 9  will have integer-valued interior angle measures

Also..... when n  = 8, the interior angle will measure  180 * 3/4  = 135°

So.... only the polygon with a side of 7 will not have an interior angle with an integer-valued measure

(4/3)x^2  + 4x + 1       factor out the 4/3

(4/3)  [ x^2  + 3x   +  3/4 ]       complete the square  on the quadratic inside the brackets

(4/3)  [ x^2  + 3x + 9/4  + 3/4  - 9/4 ]     factor the first three terms and simplify the rest

(4/3) [ (x + 3/2)^2  - 3/2 ]     distribute the (4/3)  over the terms

(4/3) ( x + 3/2)^2    - 2    =   (4/3) ( x + 3/2)^2  + (-2) 

So...in the form   a( x +b)^2 + c

a = (4/3)   b = (3/2)   and  c  = -2

So

abc  =  ( 4/3)(3/2)(-2)   =  -24 / 6   =   -4

log_a s  = 5    log_a t   =10

In exponential form, we have

a^5  = s       and  a^10  =  t

So

log_a (s / t^2 )  =

log_a [a^5 / (a^10)^2 ]  =

log_a [ a^5 / a^20 ]  =

log_a  a^(-15)        and by a log property we can write

-15 * log_a a  =

 ( Note  log_a a   =  1  )

-15 * 1  =

-15

Sorry...I missed the last point.....let me delete my answer and start over....!!

Since the base of triangle  ACB  is a diameter, then  ACB  is a right angle

And we can use the Pythagorean Theorem to find  AC

AC =  √[AB^2  - BC^2]  =  √[4^2  - 2^2 ] =  √12  units^2 =  2√3  cm

And the area  of triangle ABC  =  1/2 the product of the leg lengths  =

1/2 * AC * BC  =  1/2 (2√3) ( 2)  =   2√3 cm^2   (1) 

And the area  of the circle  = pi  * (diameter /2)^2 = pi * (4/2)^2  = pi * 2^2  = 4 pi cm^2  (2)

So....the shaded area  =

area of the circle - area of triangle ABC  =

(2)  - (1)  =

[ 4pi - 2√3 ) cm^2   ≈     9.10  cm^2

Here's one way to do this.....construct  triangles BFA , BGC, HDC and EFD

And the area of the pentagon  =

Area  of rectangle FGHI  - area  of triangle BFA  - area of triangle BGC  - area of triangle HDC - area  of triangle EFD  =

FG * GH  - (1/2)AF * BF  - (1/2) CG * BG - (1/2) CH * DH  - (1/2) EI * DI  =

8*9  - (1/2) 2 * 5  -  (1/2) 4 *3  - (1/2) 5 * 2 - (1/2) 3 * 6 =

72  - 5  -  6  -  5  - 9  =

72  -  ( 5 + 6 + 5 + 9)  =

72  - ( 25)

47  units^2

Sorry, ACG...I don't know the answer to  that one....!!!

Two  sides  of the decking  wil be rectangles with the dimensions [ (150 + 6) * 3]

And the other two sides will be rectangles  with the dimensions  ( 50 * 3)

So  the total area of the decking is

2 [  (150 + 6) * 3 ]  +  2 (50 * 3)  =

2 ( 156 * 3)  + 2 ( 150)  =

2 ( 468)  + 2 (150)  =

2 ( 468 +150)  =

2 ( 618)  =

1236  ft^2