ElectricPavlov

You can use the cosine law

d^2 = 4^2 + 6^2 - 2 (4)(6) cos 120                 (120 degrees is the angle between the two hiking segments)

d^2 = 76  

            d= sqrt 76 =8.72 miles from start

First clean up your equations by expanding and simplifying

4m-4n=7-7n

4m +3n = 7  

3/2 n = 4 - 2m

4m + 3n = 8

Both the same equation but they equal different constants.... there is NO SOLUTION....prallel lines do not cross/intersect.

Area of a rhombus =  (diagonal1 * diagonal2)/2  = 108 ft²

Rhombus diagonals bisect each other at right angles

Use pythagorean theorem to find   1/2   of the other diagonal = sqrt 2072      TOTAL diagonal length = 2 sqrt (2072) = 91.04 units

y = 5x^2-20x+1357+x^2-4x =

y = 6x^2 - 24x + 1357               will have a minimum value (y) at th4e 'x' value given by   -b /2a    = 24/12 = 2

Use this value of x in the equation to find the minimum value = 1333

9x^2 - 30x - 42 - x^2 - 2x  =

8x^2 - 32x - 42  = 0     

8 (x^2 - 4x)        - 42   = 0          'Complete the square'  for 'x'  

8 ( x-2)²    - 32    -42    = 0

8 (x-2)² - 74   = 0                           <====== Now you can see a    b   and   c   to finish

The interior angle sum to   ( n-2) * 180 = 1440

36x + 1188 = 1440

x = 7                                      so the two angles with 'x' in them are 147^o     105^o        so the smallest angle is 105^o

Kinda messy, but here is my solution

w = width

s = side

Sum of edges =  4 w   +   8 s   = 60

                              w +      2s = 15

                                   w = 15 - 2s

Surface area =  s*s    + s*s   + w*s  + w*s + w*s + w*s   =  2s^2 + 4 ws = 56   

                                                                                             s^2 + 2ws = 28        sub in the red equation for w

                                   s^2 +2s (15-2s) = 28

                                  s^2 + 30s - 4s^2  - 28 = 0

                                    -3s^2 + 30s-28 = 0               Quadratic formula show s = 1.04189    or  8.95811 (throw out ....too big)

                                                                                                             then w = 15-2s = 12.91622

Now we have the dimensions of the 3-d solid      diagonal = sqrt ( s^2 + s^2 + w^2) =~~ 13 units

Rhombus sides are all equal    148 / 4 = 37 per side

   the diagonals of a rhombus intersect at right angles (see image below)  and bisect each other

        using pythag theorem you can find 1/2 of the other diagonal to be sqrt 1173  for total digonal length 2 * sqrt 1173

Area of a rhombus = diagonal 1  * diagonal 2   / 2        = 28 * 2 sqrt1173 / 2   units²     <====== I trust you can finish

removing 1/9 from A   leaves 8/9 A   left in A
 

8/9 A  / (B + 1/9 A)   = 4/7            Then   given that   B+ 1/9 A  =  42 + 8/9 A     sub this in for  ( B +  1/9 A)

8/9 A / (42 +8/9 A) = 4/7               

8/9 A = 4/7 ( 42 + 8/9 A)                         reults in A = 63    then because  B+ 1/9 A  =  42 + 8/9 A        B = 91 marbles originally