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Arc Length = [90 x pi/180 x R] =[60 x pi/180 x W], solve for R/W
R/W = 2/3 - ratio of the area of circle R to circle W

Thanks so much! Sorry, these type of questions are difficult for me.

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

I put your answer into my online program, and unfortunately it was incorrect. I do however get one more chance to complete the problem. 

[Area of the swimming pool + deck] - [Area of the swimming pool only] = Area of the deck.

[ 156            x                56                     ] - [   150              x                50            ] =

                  8,736                                       -                      7,500                               =1,236 feet^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

Uh, any help?

Sum =N/2*[2F + (N - 1) * D, where N=Number of terms, F=First term, D=Common difference.

First term =101/200, Common difference =1/100, Number of terms =100

Sum =100/2*[(2*101/200) + (100 - 1) * 1/100]

Sum =50 * [(202/200) + (99) * 1/100]

Sum = 50 * [1.01 + 0.99]

Sum =50 *     2

Sum=100 - sum of the 1st. sequence

Second sequence = 301/200 =F, Common difference =1/100, Number of terms =100

Sum = 100/2 * [(2*301/200) + (99) * 1/100]

Sum = 50 * [(602/200) + 0.99]

Sum = 50 * [3.01 + 0.99]

Sum = 50 *       4

Sum = 200 - sum of the 2nd. sequence.

So, a(2) - a(1) =(103/200 - 101/200) =1/100

Here are a couple of candidates:

a =12 and  b=54

GCD[12, 54] =6

LCM[12, 54] =108

Also a =42 and b= 24 will give the same GCD of 6, but the LCM[42, 24] =168 which is > than the first one.

Comments on the Stars & Bars Method

The Stars and Bars method solves this directly.  It’s quite cool, because it’s not intuitively obvious that it would.   Why does the sum of a series of the relatively simple S&B, (which is just nCr with adjusted parameters), equal the solution of a bifurcated solution process, where the Partitions by decomposition of a number into all sets, and then summing the permutation count of each partitioned set? 

Though it’s not intuitively obvious, it becomes obvious (sort of) after dissecting the equations. This is like finding Wally. It may take a while to find him, but once you do then you will (usually) always see him. There are many equations and formulas like this, especially in combinatorics.

Math texts will sometimes introduce procedures requiring the use of two or more formulas and then give an already introduced formula as the direct solution. “We leave it to the student to solve the relation.”  Lancelot Link did this to me all the time.  .

Occasionally I could solve them, but usually not.  Lancelot would usually explain these relationships, but sometimes I would get this as a response: “You know I never figured that one out. I was hoping you would because I’m very curious. 

You might now see why it was the School of Mathematics and Trolling.

GA

Let the boat's speed = S
Let the distance travelled = D  [one way]
D =Speed x Time
4[S + 6] = 10[S - 6], solve for S. Distance travelled downstream = Distance travelled upstream
S = 14 kp/h - the boat's speed
D =4[14 + 6] = 80 km - distance that the boat travelled[One Way].

Oh I figured it out!  Since there are 12 pentagons (12 x 5 = 60) there must be 20 hexagons (20 x 6 = 120) so together there are 180 edges but since it is a 3D object we divide that in half for a total of 90 sides.

Let the amount deposited in the first Bank =M

The amount deposited in the second Bank =250,000 - M

0.075M + 0.0425[250,000 - M] =15,500

0.075M + 10,625 - 0.0425M =15,500

0.0325M =15,500 - 10,625

0.0325M = 4,875

M =150,000 - money deposited in first Bank account @ 7.5%

250,000 - 150,000 =100,000 - money deposited in the second Bank account @ 4.25%

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

Sorry about that!.

These price values are in dollars:

price after  0  years   =   200

price after  1  year    =   200  +  8% of 200   =   200  +  0.08(200)  =  200(1 + 0.08)  =  200(1.08)

price after  2  years   =   200(1.08)(1.08)

price after  3  years   =   200(1.08)(1.08)(1.08)  =  200(1.08)³

price after  x  years   =   200(1.08)^x

price after  20  years  =  200(1.08)²⁰   ≈   932.19

The price after 20 years is about  $932.19 .

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

Thank you!  Any tips for the second one?

One diagonal bisects  the other

a  =  √ [ 15^2  - 9^2 ]  =  √ [225  - 81 ] =  √ 144   = 12

b =  √ [ 40^2  + 9^2 ]  =  √1681  = 41

c = 9

d = 15

e = b =  41

Here's 1

Angle  ABC  =  180  - 56 -  53  =  180  - 109  =  71°

Using the Law of Sines

11/  sin ABC  =  BC / sin BAC

11 / sin 71  =  BC  / sin 53      multiply both sides by sin 53

11 sin 53  / sin 71  = BC ≈   9.29

Triangle PST  is isosceles with PT  = ST

And in a triangle with equal sides, the angles opposite those sides are also equal

So  angle (PST)   = 62

And the sum of the interior angles in triangle PST   = 180

So......

angle (PST)  + angle ( SPT) + angle (PTS)  = 180

62     +     62   +   angle (PTS)  = 180

124   +  angle (PTS)  =  180      subtract 124 from both sides

angle (PTS)  =  56

The   ratio of the surface area  of the larger solid to the smaller  is   175 /112  = 25/16

Since we are talking about surface area, we need to take the square root of this to find the scale factor of the larger solid to the smaller

So   √[ 25 /16 ]  =    5/4

So....the corresponding side of the larger solid  is

(Side length of the smaller solid) *  (Scale Factor)  =

(20)   *  (5/4)  =

20 * 5   /  4  =

100 / 4  =

25 inches