CPhill

W                            X

Z      A        B          Y

         4             2

Triangle  ZWB  is a 30-60-90  right triangle

Angle ZWB  = 60

Angle WBZ = 30

So WZ = BZ/sqrt 3  = 4 /sqrt 3

Area of WXYZ =  ZY * WZ  =  6 * 4/sqrt 3  =  24/sqrt 3 =  8 sqrt 3

inradius = area  / semi-perimeter

Let AB = BC  = x

semi-perimeter  = [ 20 + 2x ] / 2 =  10 + x

area =  sqrt [ (10 + x) (10 + x -20) (10 +x -x)^2]  =  sqrt [ (10+x) (x -10) (100) ]

5 =   sqrt [ (10+x) (x -10) (100) ] / [ 10 +x]

5 [10 + x] = sqrt [ 100 ( x^2 - 100) ] 

5 [ 10 + x] = 10 sqrt [ (x^2 - 100) ]

[10 + x] = 2 sqrt [x^2 -100]        square both sides

x^2 + 20x + 100  =  4x^2 - 400

3x^2 - 20x - 500 =  0

x = 50/3

Perimeter  =  2 (50/3) + 20  =  160 / 3

Here's a similar problem :  https://web2.0calc.com/questions/help-pls_105

The resuting  solid will equal the difference  in volumes of the cones formed by the rotation of ADC - the rotation of  ADB   =

pi/3 * AD^2  *( DC - DB)  = 

pi/3  * 7.94^2 * ( 9 - 1)  ≈  528.15

1

sin theta = 12 / 13

cos theta = -5 / 13

tan theta =  -12 / 5

               A

              40

10              

B45           D             C

Angle C =  190 - 45 - 40  = 95

Law of Sines

10 / sin 95 =  AC / sin 45

AC = 10 * sin 45  /sin 95  ≈ 7.1

[ABC ] = (1/2) 10 *  7.1 * sin 40   ≈   22.8

6000 / 2000 *  40  =  120 min

Law of Cosines

[ 23^2 - 25^2 -18^2] / [ -2 * 25 * 18 ] = cos (angle Q)  = 7/15

Once more  →  QM = 12.5

PM =   sqrt [18^2 + 12.5^2  - 2 [ 18 * 12.5] * [ 7/15]  ]  ≈ 16.44

Here's my attempt...

1 5 x x x x  and the other numbers can  be arranged in 4!  ways

1 6 x x x x  and the other numbers can be arranged in 4! ways

x x x x 5 1  and the other numbers can be arranged in 4! ways

x x x x 6 1 and the other numbers can be arranged in 4! ways

5 1 6 x x x   ....the  5 1 6  can occupy any of 4 positions and the other numbers can be arranged in 3! ways

6 1 5  x x x  ... the  6 1 5  can occupy  any of 4 positions and the other numbers can be arranged in 3! ways

So 

4 *4!   + 2* 4*3!   =   144 ways

We have the vertex form

y = a(x - h)^2  + k         where the vertex  (h,k)  = (3,1)

We  know that  (0,7)  is on the graph..so....

7  = a (0 - 3)^2  + 1

6 =  9a

a = 6/9  = 2/3

So

y = (2/3) ( x - 3)^2 + 1

y = (2/3) (x^2 - 6x + 9) + 1

y = (2/3)x^2 - 4x +6 + 1

y = (2/3)x^2 - 4x + 7

a*b*c =  (2/3) (-4) (7)  =  -56 / 3