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ohh... i forgot about 30-60-9, lol!

Angle SAD  must  = 30°

So...triangle  SAD  is a 30-60-90  right triangle....and if DS  =  1 ....then  AD  =  sqrt (3) * DS  = sqrt (3)

And  AS=  2

And angle AST  =  120°   And angle TAS  = 30°   so angle ATS  = 30°

So....AS  = TS   = 2

So  CD   =  CT  + TS + SD   =   [ 2sqrt (3) - 3 ]  + [ 2 ] + [1]    =  2sqrt (3)

So  [ ABCD ]  =   AD * CD  =  2sqrt (3)   *  sqrt (3)   =   6 units^2

Thans Guest,

I see an arrow on the top users board, which is cerainly a way of finding people.

But I do not see any arrow on the search function.   So I am a bit confused by what you have said.

You use the parallelogram-shaped sponge to create the T-shirt design. The area of the design is 66 square inches. How many times do you use the sponge to create the design?

See the answer here:  http://www.90thkilmacudscouts.com/maths/square_point1.html

not correct, sorry.

I'm getting 6*sqrt(3) - 6.

Could you post the method that you used to solve this, VooFIX  ??? 

So

We have that

3A  : 4(90 - A)  : (180 - A ) / 2   =     3   :  14   :    4

So

3A / [ 4 (90 -  A )]  =   3 /14

14* 3A   =   3 * 4 (90 - A)

42A  = 1080 - 12A

54A  = 1080

A  = 20°

The complememt of A  = 70°

Total number of  possible sets  =  C(28,4)  =  20, 475

Total number of sets  that include 2 of the top 3 students  = C(3,2) * C(25, 2)  = 3*300  = 900

Total number of sets that include all three top students  = C(3,3) * C(25, 1) =  1 * 25 =  25

So....the probability that we choose at least 2 of the top 3 students  = 

[900 + 25 ]  / 20,475  =   925/ 20475  =    37 / 819

Consider one specific triangle ABC with A=(4,8), B=(2,-6), and C=(-4,4).

The midpoint of  AB  =  [ (2 + 4) / 2 , ( 8  - 6) / 2  ]  =  (3, 1)

The slope of the line drawn between  C  and this point is  [ 4 - 1] / [ -4-3]  = 3/-7 = (-3/7)

So....the equation of the line joining these two points is

y  =(-3/7) ( x - 3) + 1

y = (-3/7)x + 9/7 + 1

y = (-3/7)x + 16/7       (1)

Likewise....the midpoint  of BC  = [ (-4 + 2) / 2,  (-6 + 4) / 2 ] =   ( -1, -1)

And the slope of the line joining A  and this point is   [ 8 - - 1 ] / [ 4 - - 1 ]  =  9/5

So....the line joining these two points is given by :

y  = (9/5)(x - 4) + 8

y = (9/5)x - 36/5 + 40/5

y = (9/5)x + 4/5      (2)

Setting (1)  and (2)  equal  we can find the x coordinate of the intersection point as

(-3/7)x + 16/7  =  (9/5)x + 4/5

16/7 - 4/5  =  (9/5 + 3/7)x

52/35  = 78/35x

52 = 78x

x = 52/78  =  2/3

And the y coordinate is  (9/5)(2/3) + 4/5  = 18/15 + 4/5   =  6/5 + 4/5  = 10/5   = 2

Here is a pic :

The answer is (5,2).

well, why don't they just actually do it themselves....

x should be 18, if u need an explanation I can do one. 

x=18

Everyone wants answers to their Aops Academy Homework, and its due very soon. XD 

Im lazy XD, its \(6\sqrt2\)

Yep.....I realized later that I made an error ....since corrected!!!!...THX!!!!

Thanks a lot for the help. 

Nailed it.

Well it's a tangent to the original equation, so I think that it's definitely correct.  

bruh don't be lazy.......that's not good. 
it's an easy problem, work it out and if you still need help then ask 

Look at the graph.....what do you think?

https://www.desmos.com/calculator/31job5ypvn

ah, i remember the AMC's lol i used to do them many years ago...

usually what i'd say is to just quickly overview the problems, read them and attack the easy ones with the less work needed first so you get those out of the way, and then you do the "harder" ones which require more work and so you may have some time remaining to check your answers and work for silly mistakes ! 

and also another thing is to stay EXTREMELY organized when like writing your steps out, so that you can go back and review it and see oh shoot i made a mistake here or wow i did it right yay! (i say this from past experiences lol when something is time limited my work tends to look like heiroglyphs after a while) 

also USE these 2 weeks to continuosly do practice problems, i'd say search up past problems or something and just do it so much that eventually when you just look at the problem you know exactly what to do without spending much time to try to figure it out :)

and btw...... Good Luck CU!! i know you will do great :D

Ah, I get it now.  Thank you.  So the equation for part b) is y-7x+29=0?

It's     x^-1  (x²-5x-24)     <=====     take the derivative of THIS      ( you see there is a  term  24x^-1   you cannot discard this portion )