CPhill

https://web2.0calc.com/questions/geometry_8804

https://web2.0calc.com/questions/geometry_26838

       P

 5                  8

Q         M           R 

      5.5       5.5

Note that cos PMR  =  -cos PMQ

Law of Cosines

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

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

5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)

8^2 = PM^2 + 5.5^2  - 2(PM * 5.5) (- cos PMQ)

5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)

8^2 = PM^2 + 5.5^2  + 2(PM * 5.5) ( cos PMQ)        add these

89 = 2PM^2 + 60.5

[89 - 60.5] / 2  =  PM^2

PM =  sqrt [ 28.5 / 2 ]   ≈  3.775

https://web2.0calc.com/questions/triangle_21266#r1

https://web2.0calc.com/questions/right-triangles_33

https://web2.0calc.com/questions/incenter_1#r1

a - 1/a  = 1

a^2 - 1 = a

a^2 - a - 1  = 0

a^2 - a   = 1

a^2 - a + 1/4  =  1 + 1/4

(a -1/2)^2  =  5/4

a -1/2 =   sqrt (5) /2                and                             a  - 1/2 =  -sqrt (5) /2

a = [ 1 +sqrt (5)] /2                                                   a =  [ 1 -sqrt (5) ] / 2

a^2 =  [ 6 + 2sqrt 5] / 4                                            a^2  = [ 6 - 2sqrt 5 ] /4

a^2 = [3 + sqrt 5 ] / 2                                                a^2 =  [3 -sqrt 5 ] /2

a^3 = a^2 * a                                                           a^3 = a^2 * a                                  

a^3 = ( 3 + sqrt 5 )/2 * (1 + sqrt 5) /2 =                   a^3 = ( 3 -sqrt 5)/2 * (1 -sqrt 5) / 2  =

[ 8 + 4sqrt 5 )  / 4  =                                                 [ 8  - 4sqrt 5 ] / 4  =                                           

2 + sqrt 5                                                                2 - sqrt 5

3 hrs 36 min =  3.6 hrs

3 hrs 45 min = 3.75 hrs

Speed of plane =  P

Speed of Wind = 30

Distance between cities = D

Distance / Rate = Total Time  .....so...

With no wind.....

2D / P =  3.6  

D / P  = 1.8

D = 1.8 P  

With wind....

D / ( P + 30) + D / (P -30) = 3.75

1.8P ( P -30) + 1.8P ( P + 30) = 3.75 ( P -30) (P+ 30)

3.6P^2 = 3.75 (P^2 - 900)

.15P^2 =  3375

P^2 = 3375/.15

P^2 = 22500

P = 150

D = 1.8 ( 150)  =  270 miles

https://web2.0calc.com/questions/coordinates_60311

The number of problems he solves on  a  normal  day  =  p*t

On the day he drinks coffee  he solves 2pt problems....so...

(4p + 5) ( t - 3)  = 2pt

4pt +5t - 12p - 15  = 2pt

2pt +  5t - 12p - 15 =  0

2pt + 5t = 12p + 15

t =  [  12p + 15 ]  / [ 2p + 5 ] 

Note that  this makes t an integer when   p= 5  and t =  5

So.....he solves    2(5)(5) =    50 problems on the  day  he  drinks  coffee

Let P  = (x,y)

PA^2 + PB^2 + PC^2  = 3PQ^2 + k

(x -2)^2 + (y -4)^2 + ( x + 3)^2 + (y -1)^2 + (x -1)^2 + ( y - 7)^2 = 3PQ^2 + k

3x^2 + 3y^2 + 24y + 80  =  3PQ^2 + k               complete the square on y    

3x^2 + 3(y^2 + 8y + 80/3)  = 3PQ^2 + k

3x^2 + 3( y^2 + 8y + 16 + 80/3 - 16) = 3PQ^2 + k

3x^2 + 3(y + 4)^2 + 3 ( 80/3 - 48/3) = 3PQ^2 + k

3x^2 + 3(y + 4)^2  + 3 (32/3)  = 3PQ^2 + k

3 [ x^2 + (y + 4)^2 ] + 32 = 3 PQ^2 + k

P =(x, y)   Q = (0, -4)

PQ^2  = [ x^2 + ( y + 4)^2]

k = 32

Triangle HQN  similar to triangle FQG          Triangle MPF  similar to triangle GPH

HQ / HN  = FQ / FG                                       MF / PF = GH / PH                    

HQ / 1  =  FQ / 2                                           1 / PF = 2 / PH                                        

HQ/FQ   = 1/2                                                 PF / PH= 1/2 

HQ = (1/3)FH   

PF = ( 1/3)FH

PQ = FH - PF - HQ  =  FH - (1/3)FH - (1/3)FH =  (1/3)FH

PQ / FH   = (1/3)FH / FH = 1/3

https://web2.0calc.com/questions/geometry_81168