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Here's the second one

MIR  is an inscribed angle

So....since it subtends minor arc  MR, the  angular measure of this arc  is twice 18° = 36°

So......the length  of MR  is given by :

2 pi  *  radius *  (36 / 360)  =

2pi * 80  * (1  /10) =

(160 / 10) pi  =

16 pi   cm

Let the amount of sales of item A be 'a', and the amount of sales of item B be 'b'.

then since a sales for 50$, you get 50a. since b sells for 25$, you get 25b.

Therefore, since the total revenue is $1500, you have for the final equation:

50a + 25b = 1500

Draw AR

Since arc AS  is 58°....then angle  ARS  also  = 58°

Similarly...draw  RB

And since arc BT is  37°, then angle BRT  also  = 37°

So...we have quadrilateral  PARB

Since   angle SRT  = 180°  ....then angle ARB  =

180  - angle ARS  - angle BRT  =  

180  - 58  - 37  =  85°

And  angles  RAP  and RBP  = 90°

And the sum of the interior angles of a quadrilateral  = 360°

So

angle ARB + angle RAP  + angle RBP  + angle APB  = 360

85  + 90  + 90 + angle APB  = 360

265  + APB  = 360

APB  = 95°

Let the midpoint of chord AB = D
Let the center of the larger circle =O
Connect AO and BO to form triangle ABO
Draw a perpendicular line from O to D
Now, you have 2 congruent right triangles with:
Angle OAB = OBA = 45 degrees
Cos(45) = 40/BA(Radius)
BA =56.568 - Radius of the large circle
Area =(56.568)^2 x Pi =~3200 pi
Tan(45) =DO(radius of the small circle) / 40
Radius = 40
Area of the smaller circle =40^2 x pi =1600 pi
3200 pi - 1600 pi =1600 pi - area of the shaded region.

unfortunately your answer was incorrect; do you think it would be correct if I type in your non-approximated answer? 

We can find half of the area of the shaded region as follows

Area of a sector of 1/6 of a circle with a radius of 12  - Area of a triangle with sides of 12 and an included angle of 60°   =

(1/2) 12^2 ( pi/3)  - (1/2) 12^2 sin (60)  =

(1/2) 12^2  (  pi /3  - √3/2)

So...the  shaded area  is twice this  =

12^2 ( pi /3  - √3/2) units^2  ≈  26.1  units^2

The formula to find the number of diagnols in a polygon is \(\frac{n(n-3)}{2}\).

Since we have 10n for the number of diagnols, we get: \(\frac{n(n-3)}{2}=10n\).

Taking 2 to the other side, we get: \(n(n-3)=20n\).

Dividing n on both sides, we get: \(n-3=20\)

This means that \(\boxed{n=23}\).

Here's 3

The total  volume  of the  first  prism is  ( 4 * 7 * 5)  = 140 cm^3

So....the price per cubic centimeter  is   4000 cents/140 cm^3   ≈  28.6 cents / cm^3

And the volume  of the second prism is  (12 * 42 * 45)  = 22680 cm^3

So...the price of the second prism (block)  is just

22680 cm^3  *  28.6 cents / cm^3 =  648648 cents  ≈   $6486.48

If we look at the  small  right triangle on the bottom left of the  kite, it has a hypotenuse of 13 and a leg of 12....so...the height of the triangle is the other leg  and  is found as

√ [ 13^2  - 12^2 ] = √ [169 - 144]   = √25  = 5

So...by symmetry, we have two  congruent triangles comprising the top and bottom of the figure

The height of both triangles is 5 and the base is  12 + 8  = 20

So....the area of  one of the triangles is  (1/2) (5) (20)  =   50

And doubling this gives  the total area  of  100 units^2 

CPhill: The answer in 3 is wrong!!!! You don't divide cubic cm by money. You divide:

$40 / 140 =$0.2857.....per cubic cm.

$0.2857..... x 22,680 cubic cm =$6,480 - cost of  the larger block of wood.

Or: [22,680 / 140] x $40 =$6,480.

Thank you!!!!

3)  You reversed the cost!!

It should be: $40/140 x 22, 680 =$6,480 - Cost of the larger block.

1) for this to be a right triangle, the other side would have to be 12, a 5/12/13 triangle.

so 5*12*1/2 for area is 30

2) the diagonals of a polygon are n(n-3)/2 this means (n-3)/2 = 10 so n = 23

3) the volume of the first block is 140 and cost $40

40/140 money  per cm^3 is the cost

the volume of the second figure is 12*42*45 or 22680

22680 times 40/140 is $6480

4) the perimiter is 98 and two of the sides are equal.

thus we have two true equations.

if the sides that are equal, called x, and the base is called, y

then 2x+y = 98  and 35^2 + (y/2)^2 = x^2

using substitution, y=98-2x we get 35^2 + [(98-2x)/2]^2 = x^2

solving for x we get, x = 37

if x = 37 then the base is 98- 2(37) = 24

now area of triangle formula (24 * 35)/2 = 420

my bad, i'll fix #3

First, in Harry's point of view: 3 sisters and 5 brothers

And, in Harriet's point of view: 2 sisters, (3-1), 6 brothers, (5+1)

So, the product of S and B is, \(S*B=6*2=\boxed{12}\).

Take a look at this link, and see if you can get an idea:

https://math.stackexchange.com/questions/227451/number-of-distinct-centroids-of-triangles-formed-by-40-equally-spaced-points-on

More detail: Since the question is asking us, how many times does 8 go into 4, we have to convert this statement into a fraction.

Analyzing, we get: \(\frac{4}{8}=\frac{1}{2}\)

\(8\div4=\frac12\)

Thanks, Daniel....The graph doesn't  clearly show it, but 0  should also be added to  the range....I have amended my answer

Isn't there also 0? That would make the answer 6.

We have that

(1/a  + 1/b)                1

________    =         __   

   a +  b                   25

[ a + b ]

______                    1

 [  ab   ]                   ___

______    =             25

  a + b

  1                    1

___       =     ____   

 ab                 25

This implies that   ab  = 25

Since a,b  appear to  be distinct  integers, the  result would be that either  1*25  = 25 or

25 * 1    = 25

Either way, their sum is 26

Since the inscribed polygon is an octagon, then the top angle of the small right-angle triangle=

180/8 =22.50 degrees.

Cos(22.5) = 4.5 / radius

Radius =4.5/cos(22.5)

Radius =4.87

Area of circle =4.87^2 x pi

                         =~74.50

Area of the octagon=sin(22.5) x 4.5 x 4.87 x 8 =~67.10

Area of the shaded region =74.5 - 67.1 =7.4

Hectictar casually solves Collatz conjecture, nominated for Fields Medal. Quote: “What’s all the fuss? It was easy!”  News @ 7:00   

Look  at the graph here :

https://www.desmos.com/calculator/mdadvpqeu9

Note the graph  is "periodic"  with a period  of  1

It appears that in each "period,"  there are 6 values in the range   (0, 0.2, 0.4, 0.6, 0.8  and  1  )

Thanks to Daniel for spotting that  "0"  should be added to the range  !!!