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We can use  the  tangent-secant theorem here

This says that  the square  of the tangent segment  =   the length of the external part of the secant segment * the  length of the secant 

We  have  that

8^2  =  4 ( 4 +x)  simplify

64  = 16 + 4x      subtract  16 from both sides

48 = 4x     divide both sides by 4

12  =  x

PV=0; R=0.05; N=25; P=18014.11;FV =P*((1 + R)^N - 1)/ R + PV*(1 +R)^N ;print FV;PV=FV;FV=PV*(1+R)^5;print FV;N=5;PV=FV;FV =P*((1 + R)^N - 1)/ R + PV*(1 +R)^N;print FV;(75000/R)/FV

$859,761.21 - Growth of his acct for first 25 years.
$1,097,297.38 - Growth for the next 5 years.
$1,499,999.74 - Growth for the last 5 years and the final value of his fund.
$18,014.11 - His average annual contribution.
$75,000 - His annual pension income for life.

cos (arctan (2))

Let  arctan (2)  = θ

This implies that

tan θ = 2  =  2  / 1    =   y  / x

And  cos θ  =  x / r

And r =   sqrt  ( x^2  + y^2)   sqrt  (1^2 + 2^2)  = sqrt (5)

So

cos (arctan (2))  =  

cos θ  =      1  /sqrt (5)  =   sqrt (5) / 5

2u       u  cannot equal or be less than - 1/2   because    2(-1/2) = -1

-20 u   u cannot equal or be MORE  than   1/20  because  -20 ( 1/10) = -1

(-1/2, 1 /20)     edited

Note  that arc ADC  =  2*measure of angle AEC =  2*110  = 220°

So  angle ADC  intercepts  arc AEC =  360 - 220  = 140°

So  we have this relationship : 

x = (1/2) (arc ADC - arc AEC)  =  (1/2) ( 220 - 140) =  (1/2)(80)  = 40° 

OK...no prob   !!!!

Thanks CPhill

If we draw radii to the tangent points we will have a quadrilateral  with two interior angles  = 90°  and  another interior angle =  central angle = 120°

And   the interior angles of a quadrilateral  sum to 360°

So

90 + 90 + 120   +  x  =  360

300 + x  = 360

x =  360 - 300  =  60°

We can solve this

0  ≤ [ -2x + 5] / 7  <  1       multiply both sides by 7

0 ≤  -2x + 5  <   7

We have two inequalities

0 ≤ -2x + 5                            -2x + 5 < 7

               Subtract  5 from both sides

-5 ≤ -2x                                                              -2x < 2

                            Divide both sides by -2

                     Reverse the inequality sign         

5/2 ≥ x                                                                  x > - 1

So....the  solution interval for   this is    -1 < x  ≤ 5/2

We can use similar triangles to solve this....see the following image

Let AD  be  one side of the square

And let   DE  be the  other side of the square

And  using  similar riangles, we  have  that triangle  BDE is similar to triangle BAC

So

BD/ DE  =  BA/ AC

BD/ DE =  2 / 3

But DE = AD

So

BD / AD   =  2/3

BD : AD  = 2 : 3

So there are 5 equal parts of AB  and AD is 3 of them...so AD = (3/5)(AB)  = (3/5)(2) = 6/5  = side of the square

Angle BCD = 74 degrees.

Using right triangles and Pythagoras, we get that XY^2 = 412.

my butwhole b***h

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Nice, hugo   !!!!!!

Look under your first post and tells us what you actually mean! Then maybe we can help you.

That's true.

Why are you asking the same question over and over again>

Don't repost your question really fast. You just asked it an hour ago.

Lol, science can help solve riddles.

Don't really understand your question, but ALL powers of 2 up to 2019 have only even prime factors:

2, 4, 8, 16, 32, 64, 128, 256, 512 and 1024 = 10 such numbers - if that is what you meant!

Some data is missing.....

I think I remember this from high school

The Doppler effect  says   that the  pitch of the sound will  be  lower  as  the distance from the object increases

So  (A) is correct

One point  on the circle  is  (3,0)   and  the other  is  (6,6)

The center  must be   (3, b)  since the circle is tangent to the x axis

Since  the  radial distance from (3,0)  to (3, b)   =  the  same radial distance from (6,6) to ( 3,b)

So....using the square of the distances  we  have that

b^2   = ( 6-3)^2  + (6 - b)^2       

b^2  = 9  + 36 - 12b + b^2

12b = 45

b = 45/12   =  15/4  =  the radius of the circle

So  we  equation of the circle  is

(x - 3)^2  + (y - 15/4)^2 = (15/4)^2

Since (p,p)  is on the circle we have that

(p - 3)^2  + ( p - 15/4)^2  =(15/4)^2

p^2  - 6p + 9 + p^2 - (15/2)p + (15/4)^2 = (15/4)^2

2p^2  - (6 + 15/2)p  + 9  =  0

2p^2  - (27/2)p + 9  = 0

4p^2 - 27p + 18  =  0  factor  as

(4p - 3) ( p - 6)  =  0

Setting both factors to 0 and so;ving for  p we get that  p = 6   (we already know this)

or

p = (3/4)

So  (p,p)  = (3/4 , 3/4)

See the graph here  : https://www.desmos.com/calculator/hasagqw5bw