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omg thank you so much :) 

The easiest way  { in my opinion)  to do this is with a graph

We  want  three circles

One with a center of (-2, 5)  and a radius of 2

One with a center of (2,3)  and a radius of 4

One with a center of (-2, -4) and a radius of 7

See the graph of these here :   https://www.desmos.com/calculator/ah1hubg7wr

The point where the  three circles  intersect is the epicenter

As the graph shows....this point is  ( -2,3)

Don't worry, LuckyDucky.....this problem would give a lot of people trouble  !!!

thank youuu

Construct the angle bisectors of two angles of the triangle. The point where the angle bisectors meet is the center of the circle.

oops

Using casework

Case 1: 6 tall people. Then (6,0) is the only possible ordered pair.

Case 2: 5 tall people and 1 short person. Then (6,2) is the only possible ordered pair: all 6 people must be sitting next to a tall person, and 2 of the tall people are sitting next to the short person.

Case 3: 4 tall people and 2 short people. Then, if the short people sit next to each other or opposite each other, our ordered pair is (6,4); if they sit one apart, our ordered pair is (5,3).

Case 4: 3 tall people and 3 short people. Then if the arrangement is SSSTTT, SSTSTT, or TTSTSS, our ordered pair is (5,5). If it is STSTST, our ordered pair is (3,3).

Case 5: 2 tall people and 4 short people. By symmetry with Case 3, the possible pairs are (4,6) and (3,5).

Case 6: 1 tall person and 5 short people. By symmetry with Case 2, the only possible pair is (2,6).

Case 7: 6 short people. The only possible pair is (0,6).

This gives us a total of 10 ordered pairs.

This is a rather difficult problem solved here by one of our best mathematicians, heureka

https://web2.0calc.com/questions/geometry_136

[ i'd like to go through this one myself   !!!  ]

 ∫  e^r  / [ 1 + e^r]   dr  =   Ln [ 1 + e^r ] + C

The key  here is to recognize  that  the derivative of 1 + e^r   =   e^r

So  remember that  if  u = 1 + e^r  then

 ∫   u '  /  u     du   =   Ln u + C

Is that correct?

ignore this ....

Let the side length of the equilateral triangle  = S

Then the wire  left  for the square perimeter  =  12 - 3S

So...the  side of the square  =   [ 12- 3S ] / 4

So....the combined areas, A,  of both figures  can be epresented as

A = ( [ 12- 3S ] / 4 )^2 +  (√3/4 )S^2

A = (1/16) ( 9S^2 - 72S + 144)  + (√3/4)S^2

A = (9/16 + √3/4)S^2  - (72/16)S + 9

Take the derivative of this and  set to 0

A '  = (18/16 + √3/2)S  - (72/16)  =  0

A '  =  ( 9/8 + √3/2)S -  4.5  = 0

(9/8 + √3/2) S  =  4.5

S =  4.5  /  ( 9/8 + √3/2)  ≈  2,26

So...the combined area is minimized when the side of the equilateral triangle  ≈ 2.26 m

See the graph here to confirm this : https://www.desmos.com/calculator/b81ztsrsgv

x = allens grocery tab

2x = original money

(2x * 1/2) * 2/3   =   6    Solve for 'x'

PV = nRT               The Universal   (or IDEAL) gas law

V = 5 liters

n = 4.65 moles

T = 39.3 C   =   312.45 K

R = ideal gas constant =  0.082 057 338(47)L atm mol−1 K−1

Plug in these values and solve for  P (pressure)   ....in  atmospheres

We can count. 

Fix the blue side of the cube to be on top.

The 2 reds can be 

1 on bottom and 1 on the side.
1 on the side and the next adjacent to it.
1 on the side and 1 opposite it.

The greens fill up all the other sides.

All other possibilities can be rotated to one of these.

So there are 3 distinct cubes.

yup, i edited it! please help :(

(a) Show that $\frac{AD}{LK}=2

Since N and M  divide sides BA and CA in the same proportion.....then NM is parallel to BC

Thus,  triangle   NAM  is similar to triangle   BAC

And   since AN  = 1/2 of AB   and similar triangles are similar in all parts,  then the altitude of triangle NAM   =1/2 altitude of triangle BAC

But altitude of triangle NAM  = altitude of triangle NML =  KL

And the altitude of triangle BAC  = AD

So

altitude of BAC / altitude of NAM   =  2/1

Which implies that

AD / KL   = 2 /1    =   2

I am new to matrices so I might be wrong. 

Firstly we multiply a and b. For the first element (the top one on a) we multiply that by every single element in b, then we do that for the second element in a, then for the third, hence we get.

0+0+0=0

-1*2+0+-1*20=-22

10*2+0+10*20=220

Then we add all this up to give us 198.

Next we do the same process except for b. Thus,

2*4+2*1+2*30= 70

0+0+0=0

20*4+20*1+20*30=700

Then we add all that up to get 770. So as of right now we have 770 + 198.

Next we do the same process for a again but to c, hence we get

0+0+0=0

-1*4+-1*1+-1*30=-35

10*4+10*1+10*30=350

350-35=315

Now, we have 198+770+315, which gives us 1283.

I am most likely wrong :).

Can you clarify this question a bit more..(i.e.) What kind of workers??

EDIT: Why did you change your question from "How do workers in Germany get paid?" to this!!!

Anyways... The percent of people who tested positive to the nearest hundredth is 13.61%.

That means that the estimate for 11.69 million people who test positive is .1361 * 11690000 = 1, 591, 009 people

*Note: I am using the same steps as CentsLord but with different numbers...

I've seen the method CPhill used before...My maths teacher called it the "Greedy Method" which means that you take the highest power of 2 from the number and subtract. Then you subtract the largest power of two from that. So on and so forth...

I don't recall an easier method but I would like to learn it!

Actually, CentsLord....I seem to recall  that there is an easier way to do this, but I don't remember the exact method....

It says round to the nearest hundredth not thousandth. This tiny error will change your answer in part 2!! 
 

*Note, this is a hint/ suggestion, not me criticizing you