CPhill

cos RMP  = -cos QMP

Law of Cosines

PQ^2  = QM^2 + PM^2 - 2(QM * PM)cos (QMP)

PR^2 = RM^2 + PM^2 - 2(RM * PM) (-cos QMP)

18^2 = 12.5^2 + PM^2 - 2(12.5*PM)cos (QMP)

23^2 = 12.5^2 + PM^2 +2(12.5 * PM)cos(QMP)        add these

853 = 312.5 + 2PM^2

[853 -312.5] / 2   = PM^2

540.5 / 2  = PM^2

PM = sqrt [ 540.5/2 ] ≈ 16.44

Simplify as

x^2 -21x + y^2 + 13y  = 25      complete the square on x and  y

x^2 -21x + 441/4  + y^2 + 13y + 169/4 = 25 + 441/4 + 169/4

(x - 21/2)^2  + ( y + 13/2)^2  = 355/2

This is a circle centered at ( 21/2 , -13/2)  with a radius = sqrt [ 355/2]  =  sqrt [ 710] / 2

The greatest value of x =  21/2 +sqrt (710 ) / 2  =    [ 21 + sqrt 710 ] / 2

Surface area =

2(LW + WH + LH) = 15

Sum of edges  =  4 (L + W + H)  = 17  →  L + W + H = 17/4....square both sides

L^2 + W^2 + H^2 + 2 (LW + WH + LH)  =  289 / 16

L^2 + W^2 + H^2 +  15 = 289/16

L^2 + W^2 + H^2  =  289/16 - 15  =  49/16

Length of diagonal  = sqrt [ L^2 + W^2 + H^2 ] = sqrt [ 49 / 16 ]  =  7 / 4  =  1.75

Let the center =  x,y

Using the square of the distance  formula  using (0,0)  and (-3, 2)

x^2 + y^2 = r^2

(x+3)^2 + (y-2)^2 = r^2       set these equal

x^2 + y^2 =  x^2 + 6x + 9  + y^2 - 4y + 4

6x - 4y =  -13     (1)

Again  using (0,0)and ( -2,1)

x^2 + y^2  = r^2

(x + 2)^2 + (y -1)^2 = r^2    

x^2 + y^2 = x^2 + 4x + 4 + y^2 - 2y + 1

4x -2y = - 5     →  -8x + 4y = 10       (2)

Add (1) ,(2)

-2x = -3

x = 3/2  

4(3/2) - 2y = -5

6 - 2y = -5

11 =2y

y =  11/2

(x, y) = (3/2, 11/2)  = (1.5, 5.5) 

Simplify as

x^2 -16x + y^2 + 20y  = 34     complete the square on x, y

x^2 -16x + 64 + y^2 + 20y + 100 =  34 + 64 + 100

(x - 8)^2  + ( y +10)^2  =  198

This is a circle centered at ( 8 , -10)   with r^2 =  198

Area =  pi * r^2  =  198 pi

https://web2.0calc.com/questions/circles_149

sin (a)  = 3/5

(c1,c2) = ( sqrt (5^2 - 3^2), 3)  = sqrt (16), 3)  =   (4,3)

sin (a) =  4/5

(d1, d2)  = ( sqrt (5^2 - 4^2) , 4)  = (sqrt (9) , 4)  = (3, 4)

LCM  =  2^4 * 3^2 * 5 * 7 * 11* 13

2 = 2                   9  = 3^2               16 = 2^4

3 = 3                   10 = 2 * 5

4 = 2^2               11 = 11

5 = 5                    12 = 2^2 * 3

6 = 2 * 3               13 = 13

7 = 7                     14 = 7 * 2

8 = 2^3                 15 = 3 * 5

LCM = 2^4 * 3^2 * 5 * 7 * 11 * 13

# positive divisors =  (4 +1) (2 + 1) (1 + 1)^4  =  240 

Depends on the factorization of m

m =  2 * 3^8   has  (1 + 1) (8 + 1)  = 18 divisors

m^2  = [ 2 * 3^8]^2  =  2^2 * 3^16  has ( 2 + 1) (16 + 1)  =  51 divisors

But

m = 2^2 * 3^5  has (2+1)(5 + 1)  = 18 divisors

m^2  = [ 2^2 * 3^5]^2  = 2^4 * 3^10  has (4 + 1) (10 + 1)  =  55 divisors

And as  NTS pointed out, 

m = 2^2 * 3^2 * 5  has (2 + 1) (2 + 1) (1 + 1)  =18 divisors

But

(2^2 * 3^2 * 5)^2 = 2^4 * 3^4 * 5^2   has ( 4 + 1 )(4 + 1) (2 + 1)  = 75 divisors

15 * 45  = 20 * x

[15 * 45 ] / 20  = x

x = 33.75 min =  33 min 45 sec