CPhill

There may be an algebraic way to do this.....but I think it can be done graphically like this :

https://www.desmos.com/calculator/4fhytbda6x

It can be shown that some corner point [ a point in the feasible region of all the graphs] will maximize x + y   [ the objective function]

Therea are two corner points here  (-5/9, 31/9)  and (5/9 , -31/9 )   

Testing both points in the objective function,the corner point  (-5/9,  31/9)  will maxmize x + y  =

-5/9 + 31/9  =

26/9

[ This is known as linear programming ]

We can use this formula to find each new point

new point  = (old point - center)*scale factor + center

So

[ (-1, 3) - ( -2 , -2) ] * 2 + (-2, -2)  = [ 1 , 5 ] *2 + (-2,-2)  = (2, 10) + (-2, -2)  = (0, 8)

[ (-5, 2) - ( -2, -2) ] *2 + (-2, -2)  =  [ -3, , 4]*2 + (-2,-2)  = ( -6, 8 ) + (-2, -2) = (-8, 6)

[(-4, -2) - (-2, -2) ] *2 + (-2, -2)  = [ (-2, 0) ]*2 + (-2, -2)  = (-4, 0) + (-2, -2) = (-6, -2)

[ (1,1) - (-2, -2)]*2 + (-2,-2)  = [ (3, 3]*2 + (-2, -2)  = (6, 6) + (-2, -2)  = (4, 4) 

Here's a pic :

In the form   ax^3 + bx^2 + cx + d

The sum of the roots  is   -b/a

So....in this case.....  -2/1  =   -2

We have the avaiable letters

A  E  G   I   K   O   P   R   T  U   V

For the first letter, we can select either the G or K  =  2 choices

For the next letter, we can pick one of 8 letters  = 9 choices

For the next letter, can pick one of 7 letters = 8 choices

For the next letter, we can pick one of 6 letters = 7 choices

The last letter must be  a "T"

So....2 * 9 * 8 * 7  =    1008  plates

EDIT  to correct an errot

Agree again, NSS...

"A" seems correct, NSS

First one

 y  = 3x

Second one

y = x^2

We'll save  (a) for last

(b)  3 / x  =  4x + 4      this will have two real solutions  

(c)  Multiply both sides by x

3 = 4x^2 + 4x       rearrange as

4x^2 + 4x - 3   = 0      factor

(2x - 1) (2x + 3)  = 0

Set both factors to 0 and solve for x and we get   x = 1/2   or  x = -3/2

When x  = 1/2 ...... y = 3 / (1/2)  = 6

When x = -3/2.......y = 3 / (-3/2)  = -2

So..intersection pts  are  (1/2, 6)  and ( -3/2, -2)

Here's the graph : https://www.desmos.com/calculator/omkotiywia

Angles DAB  and CAB  are supplementary  [sum to 180]

By the Exterior Angle Theorem,  angle DAB =   angle ABC + angle ACB

Substituting...we have that

angle ABC + angle ACB  + angle CAB  = 180

And these are the 3  interior angles in the triangle

Second one.....

Using the first answer...we know that

CA / BD  = 48/27

CA /36 = 48/27

CA/36 = 16/9

CA =  16 * 36 / 9

CA = 16 * 4

CA  = 64