CPhill

https://web2.0calc.com/questions/quadratic_84013

We can use the Law of Cosines

BC^2 = AB^2 + AC^2  - 2 ( AB * AC) * cos (120)

(x + 4)^2  = x^2  + (x + 2)^2  - 2( x * ( x + 2)) (- 1/2)    simplify

x^2 + 8x + 16 =  x^2 + x^2 + 4x + 4  + x^2 + 2x

x^2 + 8x + 16  = 3x^2 + 6x  + 4

2x^2 - 2x - 12 =  0

x^2 -x - 6  =  0

(x -3) ( x + 2)   =  0

X -3   = 0

x = 3

https://web2.0calc.com/questions/angle-problem_3#r1

3 hrs 36 min =   3 + 3/5  hrs  = 18/5 hrs

3hrs 45 min = 3 + 3/4 hrs =  15/4 hrs

Let the distance between the towns = D

So....the round trip  = 2D

Let the speed of the plane in still air = S

With the wind the the plane travels at S + 30  mph

Against the wind the plane travels at S - 30 mph

So...in still air we have this equation using Distance / Rate = Time

2D / S =  18/5

2D = (18/5)S

S = (10/18)D   =  (5/9)D   (1)

With the wind blowing we have this 

D / (S + 30)   +  D/ (S -30)  =  15/4      (2)

Sub (1)  into (2)

D / (5/9 D + 30)  + D /( 5/9 D - 30) = 15/4     simplify

D(5/9D -30)  + D(5/9D + 30)  = (15/4) (5/9D + 30) (5/9D -30)

(10/9)D^2  = (15/4) ( 25/81 D - 900)

(10/9)D^2  =  (125/108)D^2 -3375

(125/108 - 10/9)D^2  = 3375

(125 / 108  - 120 /108)D^2  = 3375

(5/108)D^2  = 3375

D^2  =  108 / 5 *  3375

D^2  = 72900          take the square root

D = 270  miles

Let AE = "a"

Note that angle AGE = 90   because GE is perpendicular to AF

Let angle  GAE =  theta

Because AF is parallel to EC , then angle GAE  =  CED

sin ( GAE)  =  1/a

tan (CED)  = 8 / (10 - a)

sin (CED)  =   8 / sqrt [ 8^2 + (10-a)^2]  =  8 /sqrt [ a^2 -20a + 164]

sin (GAE)  = sin (CED)

1/a = 8 / sqrt [a^2 -20a + 164 ]       square both sides and  cross-multiply

a^2 -20a + 164  =  64a^2

63a^2 + 20a - 164 = 0

a ≈ 1.4625

The strip is a parallelogram with a height of 8 and a width of a = 1.4625

Area =  8 * 1.4625 =  11.7 

Note something interesting.......the numerical value of the area ≈ the numerical value of the length of  AF, EC

-1 < 2x < 12    divide through by 2                       -1 < 3x - 7 < 12      add 7 to each

-1/2 < x < 6                                                           6  <   3x < 19      divide through by 3

                                                                              2 <  x   < 19 / 3

Most restrictive interval

2  <   x  <  6

https://web2.0calc.com/questions/bisectors_25

https://web2.0calc.com/questions/right-triangle_44

Break this into two parts

1 - 3x < x + 8                   x + 8 ≤  7 - x

-7 < 4x                             2x ≤  -1

x > -7/4                              x ≤ -1/2

( -7/4 , -1/2 ] 

-2 ( 6 + 2w) ≤ -16 + 7w               -3w ≥ 18 -5w + 3

-12 - 4w  ≤ -16 + 7w                    2w ≥  - 15

4  ≤ 11w                                        w ≥ -15/2

w ≥ 4/11

The most rstrictive interval  is   w ≥ 4/11