CPhill

5x - 2y = 7       (1)

3x + y = 2   ⇒  y = 2 - 3x       (2)

Sub (2)  into (1)

5x - 2 ( 2 - 3x)  =  7

5x  - 4 + 6x  =  7

11x  -  4  =  7

11x  =  7 + 4

11x  =  11

x  =1             and   y = 2 - 3(1)  =  -1

24x  -14y   =   

24 (1)  - 14 (-1)  = 

24  +  14   = 

38

We know an  angle, an opposite side and we're looking for the adjacent side

tan (angle) =   opp  / adj   ....so...

tan 22°  =  10  / x      rearrange  as

x =  10  / tan 22° ≈  24.8 km

Triangles PQS and RQS  are  under  the same height.....and triangles  under  the same height are to each other  as  their bases

So   area of  triangle  RQS  = 2 * the  area of triangle PQS

So

[ PQS ] + [ RQS]   = area of  trapezoid....so.....

[PQS ] + 2 [ PQS ]  =   32

3 [ PQS ]  = 32

[ PQS ] = 32 /  3

Let us suppose  that  the total  amount we  wish to make is  100 liters

Then  let the  amt of liquid from the 15% bottle  = x  and the  amt from the 20% bottle  = 100  - x

So...we  have

.15x  + .20 (100  - x)  =   .16 (100)

.15x  +20 - .20x  =  16

-.05x  + .20  = 16

-.05x  = .16  -.20

-.05x  = -. 4

x  =  80

100 - x  = 20

X :  Y  =   80  ;  20  =   4 : 1

We will have  a right  triangle  that is  formed  by a hypotenuse of  4001.2  miles, a leg  of  4000 miles  and   the  other  leg  = sqrt [ 4001.2^2   - 4000^2]...(this  leg  =  the  greatest distance you can see from the  balloon to the point  where this leg is tangent to the Earth's  circumference )

The arc  measure  can  be  found  as

arctan  ( sqrt ( 4001.2^2 - 4000^2 )  /  4000)  ≈  1.4°

[This doesn't sond like much because  your  height above  the Earth's surface isn't very great  ]

Arc YZ  =   360  - 90 - 90  - 60   =  120°

arc EFD  = 180  - 40 - 35 =   180  -  75   =  105°

Area of rectangle=  3 * 5  = 15 cm^2

Area of the trapezoid   = (1/2)height * (sum of base lengths)  = (1/2) (7 - 3)  (5 + 10)  = 30 cm ^2

Total area  =    (15 + 30) = 45 cm^2

Dimensions

       L          W            H

(30 - 2x)  ( 20 - 2x)   (x)

We  can slpit the paralllelogram in half  such  that  we form  two isosceles  triangles    with  the  apex angle = 160°

The  sides  are all equal  and  we  can find  the  length  of  one (= s)  as

2 / sin 20°  =  s / sin 90

s =  2 /sin 20°

Area = 

s^2 * sin (160°) =     

( 2 / sin 20°)^2  * sin (160°)   =   11.7  in^2