All Answers 358077 Answers

so you're asking 16=4^x

16=4^2 so x=2

we need to start by doing this: 1-0.4=0.6

then, 72/0.6=120

how many are not cows? 120-72=48

120+48=168

20 students

2y=-16x+10

Original radius 10 cm    now with   (360-144)/360 * (10)  radius of the cone = 6 cm

slant height = 10 cm

cone height =  sqrt(10^2 - 6^2) = 8 cm

cone volume = 1/3  pi r^2 h

                     = 1/3 pi (6²)(8) cm³ = .............. cm³

Well i'm not sure what that is, and also, 2+i seems to be wrong.

M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 10, and PR = 15, then find OR.

-------------------------------------------------------------

By the median theorem ON = QN / 3 = 10/3

NR = PR / 2 = 7.5

OR = sqrt{( 10/3)² + 7.5²} = 8.207381501

Jeff, stop asking AoPS questions.  You know it is cheating and it gives our forum a bad name.

Let G  be located on AC  such that  FG is   parallel to   CD

angle GFB  = angle CEB

BF  = BE

And angle CBE  = angle  FBG

So by ASA  triangle FBG  is congruent  to triangle EBC

So  BG  = BC

But since    AF   =  AD   then   GA  also =  GC

But  BC  +  BG   =    GC

And since BG  = BC  then      2BC  =  GC

So 2BC also   =  GA

And  AC =  GC + GA     =    2BC  + 2BC   =  4BC

And AB =   4BC  - BC   = 3BC

So

AB : BC    =   3 : 1            { just as the  guest found !!! }

We just  have a right triangle

The  hypotenuse = 34

One leg =  30

Height  = the other leg   =    sqrt ( 34^2  - 30^2)  =  sqrt  ( 256)   =    16  in

x +  sqrt y =   27

y +  sqrt x  =   9

Subtract these

x - y   +  ( sqrt y - sqrt x)   =  18                {    factor x - y    as a difference  of roots  }

(sqrt x +  sqrt y)  ( sqrt x - sqrt y)  - (sqrt x - sqrt y)    =  18        { factor out  sqrt x  - sqrt y }

(sqrt x  - sqrt y)  ( sqrt x + sqrt y  - 1)     =  18

Note  that    factors  of 18   are    1   2   3    6   9    18

And note that  

(3)  (6)  = 18

(sqrt 25   - sqrt 4 )  ( sqrt 25  + sqrt 4  - 1)  = 18

(5  - 2)  ( 5  + 2   -1)  = 18

(3) ( 6)  =  18

So

x = 25     y = 4

x + y  =   29

"c"  must be  the  value of the discriminant

Thus

5^2  - 4(1) (c)   =  c

25  - 4c  = c

25   =  5c

c = 5

Check for yourself that     x^ 2  + 5x +  5  =  0       has roots  of the  form   given

1. Just divide by 2 to get y=-8x+10 or 8x+y=10

2. 2x+y=6

3. -6y=30-5x, 6y=5x-30, y=(5/6)x-5

4.2y=-4-3x, y=(-3/2)x-2

5. first multiply by 2 to get integers, 2y=3x+40, 2y-3x=40

Hope this helped :)

cos A  +  cos^2A  = 1

cos^2A  =  1  - cos A

(1  - sin^2A)  =1 - cos A

sin^2 A = cos A                   square both sides

sIn^4A  = cos^2 A

Therefore

cos A  +  cos^2 A  =  1

So

sin^2 A   +   sin^4 A   = 1

First, square both sides

2x + 1   + 2sqrt [ ( 2x + 1)(x  - 3) ]  + x  - 3 =   4x     simplify

3x - 2 +  2sqrt [ ( 2x + 1)(x - 3)  ]  =  4x

2sqrt  [  ( 2x + 1) ( x  -3) ]   =  x + 2        square both sides again

4  ( 2x + 1) ( x - 3)  = x^2 + 4x + 4

4 ( 2x^2 -5x - 3)  = x^2  + 4x + 4

8x^ 2  - 20x - 12  = x^2  + 4x + 4

7x^2 - 24x  - 16  =   0      factor as

(7x  + 4 ) ( x  -4  )   =  0

The first factor  set to 0 and solved for x   gives  x =  -4/7

This is no good  because  we get  negatives  under  all the radicals with this solution

The second factor set  to 0 and solved for x gives us the  solution   x  = 4

The height of the larger (white) right triangle at the  upper right   = 6

And the height of the  smaller  (white) right triangle at the  lower right  =  3

And these triangles are  siimilar

So the scale  factor  is  2:1

The legs of  the  smaller right triangle  are (1/3) (3)  =  1  and 3.....and its area =   (1/2)(3)(1)  = 3/2  = 1.5

The large  white region has the  area of   (1/2) ( 9) (3 + 9)  =   54

And the  combined area of the  squares =  9^2 + 3^2 =  90

So....the black area =   

90   -  54  - 1.5   = 

34.5  units^2

The Hexagon has 18 triangles, which each have 1/3 area, and there are 12 triangles in the light blue star, so 

\(12*\frac{1}{3}=4\)

The first even integer  is  26

The last is   426

The number of  terms =    (426 - 26)  / 2   +  1  =     200  +  1  = 201

The sum is

[ 26  + 426  ] (201/2)   =  

[ 452 ] / 2  * 201 =

45426

Have a good break Dewdrop!

In an isosceles △ABC (AC = BC) its incircle touches the side BC at point T. Given AC = 4 and AT = 2. Find the length of AB.

Triangles ABC and ABT are similar.

AT = AB = 2

\(\begin{bmatrix}y=\log _2\left(x\right)\\ 2x+y-5=0\end{bmatrix}\)

\(\begin{bmatrix}2x+\log _2\left(x\right)-5=0\end{bmatrix}\)

\(y=\log _2\left(2\right)\)

\(y=1,\:x=2\)

Sorry no - 

so far i figured out that AT is angle bisector. im having trouble using angle bisector theorem or whatever else i need to use tho.