CPhill

1.   (7⅗ of 17∕ 19) ∕ 15∕25

7 3/5  =   38/3

And  "of"  means " multiply"  ......so.....

(38/3) * ( 17/ 19)  =   (38/19) * ( 17/3)  =  2 * (17/3)  = 34 / 3

And  ( 34/3 ) / ( 15/25)   =    (34/3) * (25/15)  =  170/9  ≈   18.888   [8's repeating ]

3.     log₅ (12.5) +  log₅ (2)  =   log (12.5 *  2)  =  log₅ 25 =  2

4.    √-4    is   imaginary

Here's # 2.......we need to manipulate the Law of Cosines twice.....

We can first  find  angle RST...so we have

[ RS^2  +ST^2 - RT^2] / [ 2(RS)(ST)] = cos RST

So

[13^2 + 14^2  - 15^2] / [ 2(13) (14) ]  =  cos RST

So  

arcos  ( [13^2 + 14^2  - 15^2] / [ 2(13) (14) ] )  =  RST  ≈  67.38°

Since M is the mid-point of ST, then SM  = 7

Now manipulating the Law of Cosines again to find RM, we have that

RM  =  sqrt ( SM^2  + RS^2  - 2(SM)(RS) cos  67.38° )  

RM =  sqrt ( 7^2  + 13^2 - 2 (7)(13) cos  67.38° ) ≈ 12.166

Here's an approximate pic :

The total  possible arrangements  =  12!     

Tne math books can be arranged in 3! ways

Tne history books can be arranged in 5! ways

Tne science books can be arranged in 4! ways

And these groups can be arranged in 3!  ways

So......the probability that  each category can be kept together is

[ 3! * 5! * 4! * 3! ]  / 12!   ≈  0.000216

We want choose any  5 of the 13 females  and any 5 of the 12 males =

C(13,5) * C(12,5)  = 1019304    (1)

And the total committees possible  =   C(25, 10) = 3268760    (2)

So the probability  =  (1)  / (2)  =   1019304 / 3268760  ≈  0.311832

That's OK, X²......your explanation was better than mine....

Definitely, linear....here's the graph :

https://www.desmos.com/calculator/g6zznwsare

Each minute, 1 shirt is made by 5 machines......so 100 machines must be able to make 20 times as many in one minute = 20 shirts in one minute.....so....it must take 5 minutes for 100 machines to make 100 shirts 

cot x + 6 sin x - 2 cos x =3     we can write

cosx / sinx  +  6sin x - 2 cosx  = 3       multiply through by sin x

cos x  +  6 sin^2x  -  2cosx sin x  =  3sin x

cos x + 6sin^2x  - 2sin cos x  - 3sin x  = 0         we can factor this as

( 3sinx  - cosx) (2sinx  - 1) = 0      set each factor to 0  and solve

3sinx  - cos x  = 0  

3sin x  = cos x

3sinx / cos x  =  1

3 tan x  =  1

tan x  = 1/3 

arctan (1/3)  ≈  18.435°  + 180°n    where  n is an intreger

For the other factor we have that

2sinx - 1  = 0

2sin x  = 1

sin x  =  1/2     

arcsin (1/2)    and this happens at  30° + 360°n    and at  150° + 360°n

And again, n is an integer

Here's ther graph : https://www.desmos.com/calculator/eiai1cs0i1

tan (pi /6)  =  1 / √3   =  √3 / 3

Here's one possibility.....it's known as the greatest  interger function :

https://www.wolframalpha.com/input/?i=greatest+integer+function

It's denoted   as    shown here   : 

http://www.mathwords.com/c/ceiling_function.htm

Note

lim  f(x)     =    1        

x → 2^- 

But

lim  f(x)     =    2        

x → 2⁺