Melody

This question has been done to death!

Let the pieces be   x,  y-x  and  5-y       (that adds up to 5)

Now draw a contour probability map.

It is 3/4 * the circumference of the circle  + 2 radii.

The formula for the circumference of a circle is 2pi*r

Now you can do it.

 y=16/(5+3x)

inverse

x=16/(5-3y)

-2= 16/(5-3y)

-2(5-3y)=16

3y-5 = -8

3y = -3

y=-1

\(f^{-1}(-2)=-1\)

Find the least positive four-digit solution to the following system of congruences.

7x = 21 (mod 14)

2x + 18 = 16 (mod 9)

7x=21 mod14

x=3 mod14

x=14p+3

2x+18=16 mod9

2x=-2  mod9

x=-1    mod9

x=9q-1

9q-1 = 14p+3

9q - 14p = 4

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

9=-1*(-14) - 5

-14=  3(-5)+1

-14 - 3(-5) = 1

-14 - 3 [9+1*(-14)] = 1

(-14) - 3*9 -3(-14) = 1

-2(-14) -3(9) = 1

9(-3) -14(-2)=1

9(-12) -14(-8)=4

9(-12+14g) - 14(-8+9g) = 4

so

q =-12+14g             and    p =-8+9g

x = 9q-1                  and     x= 14p+3

x= 9(-12+14g)-1     and     x= 14(-8+9g)+3

x= -109 +9*14g      and     x = -109 + 14*9g      they are the same so it is right.

x= -109 +126g

I want the smallest 4 digit value of x

-109+126g>1000

126g > 1109

g > 8.8

g=9

x = -109+126*9

x= 1025

sketch them using Desmos

Let f(x) = 2^x.  Find sqrt(f(sqrt(f(f(1)))))

sqrt(f(sqrt(f(f(1)))))

=sqrt(f(sqrt(f(2))))

=sqrt(f(sqrt(4)))

=sqrt(f(2))

=sqrt(4)

=2

ok, sorry

You need to identify yourself next time. 

If you become a member that would be the easiest way.  

Plus if you make a good impression your questions will be answered better and quicker. 

I have not looked at your answer in any great detail but it looks fine to me.  (I don't have my solution anymore)

Guest, please do not answer over the top of me.

I give partial answers for a reason and the asker is always encouraged to ask for more help if required.

I do not want this to be just a cheat site. 

I want it to be a teaching / help site.  I think that is what most of the answerers here would want.

If you want to answer a question in full that is fine,  but choose one, of the many, that does not already have a correct partial answer.

Thank you.