Girls vs boys maths challenge

Although I did have to use your hint about the arrangement of the semicircles. Can I ask how you knew instantly that that maximised the area covered by them (i.e. is there a specific rule that applies more generally)?

Just instinct. I checked other configurations to make sure they weren't better.

Yep, exactly I guessed that the signs would just be a typo or carelessness or something - they're not something I would worry about when solving a problem, I'd just bung them in at the end and hope for the best!

dF/da = -s (s-b) (s-c) - 1/2 * lambda, right?

Yep

Do I let both derivatives equal 0 then equate?

This is because the complex logarithm is multivalued - you've implicitly taken an unusual value of log(i). (That is, in implicitly using e^log(i) = i).

Yep again

So then if I cancel out the common terms, I get s-a = s-b which just simplifies to the constraint s = (a-b)/2?

How does that simplify in that way? If $s-a = s-b$, then $-a = -b$ so $a=b$. And that's exactly what we wanted - two sides are the same length. But there was no particular reason to call this side a and this side b - we might just as well have called this side b and this side c, so with no further work we can also say that $b=c$. Hence equilateral.

For GCSE students mainly:

$tanx=\dfrac{cosx}{sinx}$ never true, sometimes true or always true?

Prove your statement

Well done .. what about the cube root of i?!

LOOOL these don't seem like GCSE q's anymore, and can people explain how they got their answer more effectively for others that do not understand


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