hectictar

Imagine that there's only  1  question. How many paths are there?

Path 1:  yes

Path 2:  no

So 1 question makes  2  paths.

If there are  2  questions, how many paths are there?

Path 1:  yes, yes

Path 2:  yes, no

Path 3:  no, yes

Path 4:  no, no

So  2  questions make  4  paths.

If there are  3  questions, how many paths are there? I'll get it started...

yes, yes, yes

yes, yes, no

yes, no, yes,

yes, no, no

no, yes, yes

. . .

That is not the complete list. Does that help any?

Remember we are given that there are 8 paths; we want to know the minimum number of questions that make 8 paths.

They're coating spherical donut holes, so...

SA of Niraek's donut hole  =  144pi  square millimeters

So to coat one donut hole, Niraek has to cover  144pi square millimeters  with powdered sugar.

In the time Niraek covers 144pi square mm, Theo will have covered 144pi sq mm because they are coating at the same rate. But 144pi sq mm for Theo is not an entire donut hole. To coat one entire donut hole, Theo has to cover 256pi sq mm.

My answer could be wrong!!

LOL!!!!

SQ : QT  =  5 : 2

x coordinate of Q   =   -2 + (5/7)(5 - -2)   =   -2 + (5/7)(7)   =   -2 + 5   =   3

y coordinate of Q   =   -6 + (5/7)(8 - -6)   =   -6 + (5/7)(14)  =  -6 + 10  =  4

point Q is located at  (3, 4)

sin of an angle  =  length of side opposite that angle / length of hypotenuse

So in this case...

sin 53.1°  =  4 / r       Multiply both sides of the equation by  r .

r sin 53.1°  =  4         Divide both sides of the equation by  sin 53.1°

r  =  4 / sin 53.1°       Plug this into a calculator to get...

r  ≈  5

x² + y² - 4x + 2y  =  b        We need to get this into standard form. Rearrange the terms.

x² - 4x   + y² + 2y   =   b        Add  4  and  1  to both sides to complete the squares on the left side.

x² - 4x + 4  + y² + 2y + 1  =  b + 4 + 1          Factor each perfect square trinomial.

(x - 2)(x - 2) + (y + 1)(y + 1)  =  b + 4 + 1

(x - 2)² + (y + 1)²   =   b + 5

Now that this is in standard form, we can see that the  y  coordinate of the center is  -1 .

And     (the radius)²  =  b + 5       So

7²  =  b + 5

49  =  b + 5

44  = b

point D  =  midpoint of AB  =  \((\frac{4+2}{2},\frac{6+-2}{2})\,=\,(\frac62,\frac42)\,=\,(3,2)\)

point E  =  midpoint of AC  =  \((\frac{4+-2}{2},\frac{6+-4}{2})\,=\,(\frac22,\frac22)\,=\,(1,1)\)

slope of DE  =  \(\frac{2-1}{3-1}\,=\,\frac12\)

slope of BC  =  \(\frac{-2--4}{2--2}\,=\,\frac24\,=\,\frac12\)

slope of DE  =  slope of BC

therefore  DE  is parallel to  BC .

If  B  is the  x-intercept and  E  is the  y-intercept of  x + 3y  =  18 , then

point B  =  (18, 0)       and       point E  =  (0, 6)

Then...

point A  =  (-18, 6+6)  =  (-18, 12)

Line M  is perpendicular to line L.

slope of line L  =  -6/18  =  -1/3   so..

slope of line M  =  3

So line M passes through the point  (-18, 12)  with a slope of  3 . So the equation of line M is...

y - 12  =  3(x - -18)

y - 12  =  3(x + 18)

y - 12  =  3x + 54

y  =  3x + 66

Ah you got it!!

I got messed up because if you try to check it in WolframAlpha by entering

cos(atan(20/21)/2)

you get a different answer, but if you enter

cos((atan(20/21)+pi)/2)

it gives the right answer.