CPhill

We have an inscribed hexagon

This will be composed of  6 equailateral  triangles  with sides  of 16

The total area  of this hexagon  =  6* (1/2) 16^2 * sin (60°)  = 768 * √3/2  = 384√3  units^2    (1)

And the circle will have a radius of 16....so its area is  pi (16)^2 = 256 pi  units^2    (2)

So...the shaded area  =  (2)  - (1)  = [ 256 pi  - 384√3]   ≈  139.1  units^2

Not as hard as  it seems

We have  a rectangle with an area  of  Width * Height  = 34 * 14  = 476  units^2     (1)

Note...that at the two ends, we have two semi-circles with a radiuses of 7 

So...the total area of these combined will actually be a circle with a radius of 7  = pi * 7^2  = 49pi units^2   (2)

So the shaded area  =  (1)  - (2)  =  476  - 49pi  ≈   322.1  units^2

We can use partial fractions, here

Note that  x^2 -1   can be factored as  ( x -1) ( x + 2)

So we have

( x + 2)                                         A         +      B

___________           =              ___             ___

(x -1) ( x + 1)                             x -1             x + 1

Multiply  through by (x -1) ( x + 1)

x + 2  = A(x + 1)  +  B(x -1)    simplify

x+2  = Ax + A  + Bx  - B

1x + 2  = (A + B)x + (A - B )     equate coefficients  to form this system

A + B  =1

A - B  = 2          

So   A  - B   = 2

This is an (irregular)  pentagon

The sum of the interior angles is  (5 -2) * 180  = 3 * 180  = 540°

So...the measure  of the missing angle is

[ 540  - 86 - 140  -138  - 59 ]  =   117  =  x 

We have  an irregular trapezoid

The top  base  = (32 + 24)  = 56 ft

The bottom base is ( 24 + 32 + 18)  =  74 ft

The height is 23 ft

So....the area  is

(1/2) height *  ( sum of the base lengths)  =

(1/2) ( 23)  ( 74 + 56)  =

(23/2) ( 130)  =

23 * 65  = 

1495  ft^2 

This is a regular heptagon (7-sided figure )

It will be composed of 7 congruent isosceles triangles.....

Their  equal sides will be 5  and the  included  angle between these  sides = (360 / 7)° 

The  area  is given by

7 ( 1/2) 5^2 *  sin  (360 / 7)°   ≈  68.4  units^2

The interior  angles of a heptagon sum to   (7 - 2) * 180   = 5 * 180  = 900°

So....the remaining angle is just

900 - 145 - 115 - 152 - 87 - 90 - 150   =   161°

The top layer  of cubes shows  ( 3 + 3 + 3 + 2)  = 11 faces

The middle layer shows ( 4 + 4 + 2  + 2)  =  12 faces

[ Note that the middle layer shows 8 faces on the front and back, 2 faces on the sides,  plus 1 face each on the top and bottom

The bottom  layer  shows  (3 + 3 + 3 + 2)  = 11 faces

So...the surface area in square  units  is  ( 11 + 12 + 11 )  =  34 units^2

How many ways are there to put 4 balls into 3 boxes, given that the balls are not distinguished but the boxes are?

Note that the boxes are distinguishable....so....(4, 0, 0)  is different from (0, 4 , 0)

The  total number of ways  =  

C ( n + k - 1, k)     where  n = the number of boxes   and  k  = the number of balls

So we have

C ( 3 +  4  - 1 , 4)  = C (6 , 4)  = 15  ways

We can confirm this by noting that

(4, 0, 0)   has  C(3,1)   =  3 arrangements

(3,1,0)  has 3!  = 6 arrangements

(2,2, 0)  has C (3,1) = 3 arrangements

(2, 2 , 1)  has C(3,1)  = 3 arrangements

So

3 + 6 + 3  + 3   =   15  arrangements (ways)

To convert  rads to degrees :

rads  *  180  / pi      ....so we have...

pi / 3  *  180  / pi  =

180 /3  =   60°

And

tan (60°)  = √3  =  tan (pi/ 3 rads )