CPhill

Draw DE , CF  perpendicular to  AB

Let AE  = x

Let BF = 96 - (44 + x)  = 52 - x

Angle DAB = 70

tan DAB  = h /AE

tan(70)  = h/ x

h = x tan 70

tan (55)  = h / (52 - x)

h = (52 -x)tan (55)

So

x tan 70  = (52 -x) tan 55

Solving  for x   ≈ 17.785

h = x tan 70  =  17.785 tan 70

Area  =  (1/2)  17.785 * tan 70 *( 96 + 44) ≈ 3420 .5

https://web2.0calc.com/questions/algebra_56153

Simplify as

x + y + 3z = 10

7x + 9y + z  = 18

kx + 14y +7z = 8

If the system has no solutions, the determinant =  0

So

1  1   3   1   1

7  9   1   7   9

k  14  7  k   14

[ 1*9*7 + 1*1 *k + 3*7 *14 ] - [ 9*3*k + 14 * 1 * 1 + 7*7*1 ]  = 0

[ 63 + k + 294] - [ 27k + 14 + 49 ] =  0

294 - 26k   = 0

k = 294/26   =147 / 13

Draw AE and BF  perpemdicular to BC

Let DE  = x

Let CF = [ 104 - ( 78 + x ) ] = 26 - x 

We have rwo right triangles ADE and BCF

Let the height of the  trapezoid  = h

So

h = sqrt [  10^2 - x^2]

h = sqrt [ 24^2 - (26  -x)^2 ]

So

sqrt [ 10^2 -x^2]  = sqrt [ 24^2 - (26 -x)^2 ]      square both sides

10^2-x^2  = 24^2 - (26 -x)^2

100 - x^2  = 576 - x^2 + 52x - 676  simplify

52x - 200 = 0

13x - 50 = 0

x = 50/13

h = sqrt [ 10^2 - (50/13)^2]  = 120/13

Area of  trapezoid   =  (1/2) (120/13) (78 + 104)   =  840

https://web2.0calc.com/questions/counting_57351

https://web2.0calc.com/questions/polynomial_8831

                      C

         E                      D

                     H

A  20          144                       B

ABH forms a triangle

Angle ABH = 180° - 144° - 20°  = 16° 

2*12^5 + 10*12^4 + 3 *12^3 + 1* 12^2 +  1*12  + 2   = 710366

11* 12^4 + 7 * 12^3 + 5 * 12^2 + 1 * 12 + 2  = 240926

710366 - 240926  = 469440

Convert back t base 12

469440  / 12  = 39120 R 0

39120 / 12 = 3260 R 0

3260 / 12 = 271 R 8

271 / 12  = 22 R 7

22 R 12  = 1 R 10

1 /12  =  0 R 1

1A7800₁₂

11₂  = 3

110₂ = 6

1000111₂ = 71

101011₂  =  43     

111₂  =   7

10111₂  =  23   

 1100101₂ = 101

3 + 6 + 71 + 43 + 7 + 23 + 101   = 254

Convert back to base 2

254 / 2  =  127  R 0

127  / 2  = 93 R 1

63 / 2  = 31 R 1

31 / 2 =   15 R 1

15 / 2  = 7 R 1

7 / 2 = 3 R 1

3 / 2 = 1 R 1

1 / 2 = 0 R 1

11111110₂

35A811 =   3 * 11^5  + 5 * 11^4 + 10 * 11^3  + 8 * 11^2  + 1 * 11 + 1   = 570648

A0411  = 10* 11^4  + 0 * 11^3  + 4*11^2 + 1 * 11  + 1   = 146906

570648 + 146906  =  717554 base 10

Convert  back to base 11

717554  / 11  = 65232 R 2

65232 /11 = 5930 R 2

5930  /11  = 539 R 1

539 / 11 = 49 R 0

49  /11  = 4 R 5

4 /11  = 0 R 4

450122