CPhill

We have an example of the ambiguous case = SSA

Using the Law of Sines

sin B / 10  = sin C / 9

sin B /10 = [sqrt (3) /2] / 9

sin B = 10 sqrt (3) /  18 = (5/9)sqrt 3 =  

arcsin [ (5/9)sqrt 3 ] = B  ≈  74.2°  or    B  ≈ 105.8°

So

So , in one case, C =  180  - 60  - 74.2   ≈ 45.8°

So...let's figure a2 first

a2 / sin (45.8)  = 9 /sin C

a2 / sin (45.8) =  9 / [ sqrt (3) / 2 ]

a2 =sin (45.8) * 18 / sqrt 3  ≈  7.45 

And in the second case

C = 180 -60 - 105.8   ≈  14.2°

So....by the Law of Sines again we can find a1 as

a1 / sin (14.2) = 18/sqrt 3

a1=  sin (14.2) * 18 /sqrt 3 ≈  2.55

{a1 , a2}  = {2.55 , 7.45}

Welcome  !!!!

Law of Sines

sin (pi/6) / (3x)  = sin B / ( x )             simplify

(1/2) / 3  = sin B

sin B  = 1/6

12 sin (B )  = 2

Since  sin B  = 1/6         then the altitude must be  1   and  x  must be 2   

Using the Law of Sines again

sin (pi /6) / (3x) =  sin (C) /AB

(1/2) /  6  =  sin (C) / AB

(1/12) = sin (C) / AB

sin C  =  AB / 12

So

12 sin C   =  AB

Law of Sines

sin ( A ) / 13  =  sin C / 24

sin ( A) / 13  = sin ( 2A) / 24

sin (A) / 13  = [ 2sin (A) cos (A)] / 24

1/13  = cos ( A  )/ 12

12/13 = cos ( A )

sin (A)  =  sqrt [ 13^2 - 12^2] / 13  =  5/13

sin (C)  =  sin (2A)  =  2sin(A)cos(A)  =  2 (5/13) (12/13)  = 120 / 169

Excellent, catmg   !!!!

sec 4x  =  csc (x - 60)

The secant and co-secant  are co-functions   ....so.....this means  that

4x + (x - 60)  = 90

5x  = 150

x = 30

We have a rectangular prism

The volume  is  (base area ) * (height) =  

( EF * EH) * (EA)  =

( 4 * 5 ) * ( 5)  = 

100 units^3

Thanks, guys   !!!!

Note that  a square is a special case of a  rectangle....but a rectangle isn't necessarily a square    

Since AB  = 4 ....then  it must also be that BC = 4

So....the area  =  (AB) (BC) =  (4)(4)  =  16

I think you might have assumed a right triangle with legs of 8 and 12....in that case, your answer would be correct