C2 Questions

With a homework titled 'Radians', I've got two annoying questions here which I could do with some help with - pointing me in the right direction would be greatly appreciated.(ii) Find the area of the triangle
(iii)Find the area of the three segments surrounding the triangle. - I know this is relatively easy once I have the area of the triangle (Part ii).

2. Once upon a time a hermit found an island shaped like a triangle with straightshores of lengths 6 km, 8 km, and 10 km. Needing seclusion, he declared that noone should approach within 1 km of his shore. What was the area of his‘exclusion’ zone?

Really not sure where to go with the latter, I've drew the triangle and 'exclusion zone' (circle), but I can't think of anything related to radians bar area of a sector. I know to do this I'll need to work out the necessary angles (I'm assuming I use the cosine rule) - but once I have the angles of the triangle, where do I then go?

With a homework titled 'Radians', I've got two annoying questions here which I could do with some help with - pointing me in the right direction would be greatly appreciated.(ii) Find the area of the triangle

no 2 is rather nice ... they should ask it in a Maths interview ...

You know that the area of a triangle is given by $\frac{1}{2}AB \sin c$

You know it's equilateral, so $c = 60^{\circ}$, both side lengths would be the same as well.

It's not bad is it, made me think so far and I'm yet to even solve it! :'
Going to give it another attempt tomorrow after 40 winks and see how far I get.

I'm assuming Cosine and trig is necessary?

Yep, I managed to get that far. The problem was with working out the side length. I got the side length to be 10√3 - but this required a very long, tedious process and I'm not too sure on it anyway.

Does this help?

An able experienced student should be able to do it without writing much down

Yep. Using that I got the same side value of 10√3 which I got using my very prolonged method! Hopefully this is right, would you possibly be able to support this answer?

Also, any idea on no. 2?

Really not sure where to go with the latter, I've drew the triangle and 'exclusion zone' (circle), but I can't think of anything related to radians bar area of a sector. I know to do this I'll need to work out the necessary angles (I'm assuming I use the cosine rule) - but once I have the angles of the triangle, where do I then go?

I like to consider myself able, although experienced I am not >.>
I'm quite looking forward to seeing the method though.

Don't you just need to find area circle - area triangle? Area triangle should be easy to work out using Heron's.

I personally can give you the answer without writing anything down ... maybe I have seen this type of problem before ....

You know what, I didn't even consider that - I'm too busy over-complicating it.
I'll give this a shot, although I have no idea what Heron's formula is!

I'm envious

Don't bother using Heron, bit of a fancy shmoozle, stick with what you do know.

I just encountered a problem: obviously I have to work out the area of the circle, but how do I go about this if I don't know the radius?

Ah, nevermind. The method you're meant to use is to notice that the excluded region consists of a rectangular region (1 km wide) across each shore and a circular region at the vertices. Picture:



That should do the trick.