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Yeah, but it's one of the possible cases, so it should be in the denom and counted.

my name jeff

You just have to copy and paste it into the address bar. Then it will work just fine :)

given cos(π/4 + x) = m

what's the value of sin2x, in terms of m?

How about:

\(cos(π/4 + x) = m\\ cos(\pi/4)cosx - sin(\pi/4)sinx=m\\ \frac{cosx}{\sqrt2}-\frac{sinx}{\sqrt2}=m\\ cosx-sinx=\sqrt2\;m\\ \text{square both sides}\\ cos^2x+sin^2x-2cosxsinx=2m^2\\ 1-2cosxsinx=2m^2\\ -2cosxsinx=2m^2-1\\ 2cosxsinx=1-2m^2\\ sin(2x)=1-2m^2 \)

Very impressive!  That looks really hard !

You made lots of mistakes in case 3, 4, 5. They simply don't add up.

Thanks Tertre   

You got most of it too    

700000 is not divisible by 77, by the way.

Good job, Melody! 

Got it  

The longest peice could be     15.5,  20.5,  17.4,  16.4,   or   21          added together this is exactly  90.8 inches

So there are 5 different possibilies on how the rope is cut.   (only lenghths are important, not the order thay are cut in)

Rather than spoil everyone's fun I will let you individually work out what these 5 double cuts  are.

Hint:   Let one peice be   a   and another   a+5.      You have to work out the other one.   (5 possibilities.)

Thanks, everyone! 

Did you use calc or use approximates?

No, actually I was trying to find the cases that would work. You could have the number be 700000.

It actually gets around 0.2500002, so 0.25.

If we squared it, though, we would get 2(sqrt(3.482)), which wouldn't simplify.

Yeah, I just approximated it.

Try to square it. Do you think approximations would help? 

(a) If you know the Pythagorean triple (5, 12, 13), the answer is 13. Thus, \(\sqrt{25+144}=\sqrt{169}=\boxed{13}.\)

Wait, so the second "7" is fixed, too? So, 1/16*5=5/16?

\(2\sqrt{1.75+\sqrt3}-(1.75+\sqrt3)\\ \)

I'm coming up blank.

but   sqrt root of 3   is irrational so it is not exactly equal to what you said.  

Case 1: 1st rope: x, 2nd rope: x+5, and the third rope 2x.

So, 4x+5=36, 4x=31, x=7.75 and 7.75*2=15.5 inches

Case 2:  1st rope: x, 2nd rope: x+5, and the third rope 2x+10.

So, 4x+15=36, 4x=21, x=21/4=5.35, and 2(10.25)=20.5 inches.

Case 3: x, 2x, 2x+5, and we have 5x+5=36, 5x=41, x=8.2 and 2(8.2)=16.4+5=21.4.

Case 4 There are other cases, maybe 2x, 4x, 4x+5 gives 10x+5=36, x=3.1 and 4(3.1)+5=17.4

This is 74.8 and there is another 16?

How about 2x, 2x+5, and 4x. 15.5?

Well, since there's no divisibility rule for 7, here's what I did:

77 can only divide the number if there are an even number of 7s. Also, the number has to start with 77 because if it only started with 7, then how would it be divisible by 77? So, we could have the numbers:

770000

770770

770077

777700

777777

There are 5 numbers. Each has a probability of (1/2)^5 of occuring, because the 7 in the hundred thousands place is already determined. So, I got 5/32, but the answer is actually 5/16? So what did I do wrong?

how did you figure this out?

Hint: A number is divisible by 77 when it is divisible by 7 and 11. There are a total of 2^6=64 total outcomes. 

I cannot see any image.  :/