CPhill

I'm not sure....but here's my best effort

Let ABCD  be the  trapezoidal field

Its  area   = 

(1/2)(height) (DC + AB) = 

(1/2)(50√3) (100 + 200) =

7500√3   m^2  

And  the  crop will be brought to the longer side whenever it is contained within  the trapezoid  AEFB

Reasoning :

DB bisects  angle CBA....angle FIB  = FKB  and FB  = FB

So....by AAS, triangle FIB  = triangle FKB

So....any crop harvested within triangle FIB will be closer to IB than BK

For the same reason any crop harvested within triangle EJA  will be closer to JA  than side AL

And any crop harvested within EFIJ  will be closer to the longest side than to side DC

And the area of AEFB  =

(1/2)(EJ) (AB + EF)  =

(1/2)(25√3) (200 + 50) =

3125√3  m^2

So  the fraction of the crop brought to the longest side  is

3125√3  / 7500√3  =   5/12

If the   quadratic function opens downward, we will have a negative sign in front of the x^2  term...so we can eliminate the last two answers right away

In the form     ax^2  + bx + c, the  x coordinate of the vertex  =  -b / [ 2a]

Look at answer B.....letting a  = -1   and b  = 4....the x coordinate of the vertex  is

-4  / [ 2 * -1 ]   =  -4 / -2   =   2

To find the y value associated with this x value, put this "2"  back into the function

-(2)^2  + 4(2) +   1   =

-4  +   8  +   1    =

-4  +  9   =

5

So....this function  turns downward  and has a vertex  of  (2, 5)

Here's a graph to confirm this  :  https://www.desmos.com/calculator/tlavaymezd

r(x)  =  1 / (1 - x)

r(30)  =  1 / ( 1 - 30)  = -1/29

r(r(30))  = r(-1/29)  =    1 / (  1 -  -1/29)  = 1 / (1/ 30)  = 30

r(r(r(30))) =  r(30)  =  1 / (1 - 30)  =  -1/29

Note  that the pattern repeats....so when r is applied 6 times we will end up with  -1/29

Thanks, Daisy...!!!

Here's a pic of the situation :

\(f(x) = \left\{ \begin{array}{cl} x+2 &\text{ if }x>3, \\ 2x+a &\text{ if }x\le 3. \end{array} \right. \)

Let x  = 3  and set the functions equal.... we have

3 + 2  = 2(3) + a

5  = 6 + a

5 - 6  = a

- 1  =  a

See the graph  here  that shows that this value of a makes the function continuous :

https://www.desmos.com/calculator/ux7xrnhaxi

1)  Note that  25^(17/x)  = ( 5^2)^( 17/x)  =  5^(34/ x)

25^(-2)  =  5^(48/x)  / [ 5^(26/x) * 5^(34/x)  ]

(5^2)^(-2)   =  5^(48/x)  /  [ 5^(60/x)]

5^(-4)  =  5^(48/x - 60/x)

5^(-4)  = 5^(-12/x)

We have the same bases, so we can solve for the exponents

-4  =  -12 /x  multiply through by x

-4x  = -12   divide both sides by -4

x  = 3

Last one

The point  (0, 3) = (x, y)  is in the shaded area

Notice  that  y - x  < 0   is not true so the first and last answers can be eliminated

And  in the third answer  y ≤ 1 is not true, either

So.....the second answer is correct

We need to find the total amount produced per year  and divide this by 365

( 2.6 x 10^2) ( 3.9 x 10^11)  / 365  =

( 10.14  x 10^13 ) / ( 3.65 x 10^2 )  =

( 10.14 / 3.65)  x (10^13 / 10^2)  ≈  2.8 x 10^11 pounds per day

Second one

2x + y   ≥   -4     rearrange as

y  ≥  -2x  - 4

This graph is a solid line  with a positive slope and a y intercept of -4

y  ≥ 2x

This is a solid line with a positive slope  going through the origin

y - 3  >  (1/2)x

y > (1/2) x + 3

This is a dashed  line with a positive slope  and a y intercept of 3

Thus the graphs on the top right and  bottom left can be eliminated since they contain lines with negative slopes

Pick the point (0, 6)  = (x , y )

Notice that this point will satisfy all three inequalities

Since this point is in the shaded area of the graph on  the bottom right  and not in the shaded area of the graph on the top left, then the graph on the bottom right must be correct

First one

y  < -(3/4)x + 1

This graph will have a negative slope and will be a  dashed line going through the y intercept of -1

y > (2/3)x  - 3

This graph will be a  dashed line with a positive slope  and  going  through the y intercept of  - 3

Thus....either graph on the top right or the one on the bottom right are correct

Pick a point  say  (0, 0)  = (x , y)

Notice that this point will satisfy both inequalities

Since this point is in the shaded region of the one on the bottom right...thus...this one is correct