First transforming the wanted inequality so its easier to solve:
Because some people want to know solutions of these questions?I like how you answer your own questions
^.I like how you answer your own questions
Post solutions that does not involve divergence/convergence? Thx.A good Sydney grammar question
I'm afraid you would have to prove the sum converges in order to find it, however I doubt that the Sydney grammar teachers who made this question expected students to do so.Post solutions that does not involve divergence/convergence? Thx.
Thx Sy123.I'm afraid you would have to prove the sum converges in order to find it, however I doubt that the Sydney grammar teachers who made this question expected students to do so.
-> A Sydney Grammar teacher may be satisfied with simply the statement that u_n is increasing therefore 1/u_n converges to zero (I could be wrong, I don't know the marking criteria, however that is my guess), however a rigorous proof of the limit is given below:
On first thoughts, you would just construct a circle within the square which has a radius a quarter that of the side length of the square. Then do the area of the circle divided by the area of the square. Similarly for the bonus, you construct a sphere and do the volume of the sphere divided by the volume of the cube.A point P is chosen inside the unit square at random. What is the probability that P is closer to the centre of the square than any side thereof?
Bonus: Solve this for a cube, replacing the word "side" with "face".
Nah, the regions won't be circles and spheres.On first thoughts, you would just construct a circle within the square which has a radius a quarter that of the side length of the square. Then do the area of the circle divided by the area of the square. Similarly for the bonus, you construct a sphere and do the volume of the sphere divided by the volume of the cube.
Will check now...I didn't actually do the calculation.I got,
I want to figure out the bonus before I post my solution for this one.
Its just 4 times the area bounded by the region (in the Cartesian plane where the square's co-ordinates are (0,0) (1,0) (1,1) and (0,1) ) , ??Will check now...I didn't actually do the calculation.
yeah that sounds about rightIts just 4 times the area bounded by the region (in the Cartesian plane where the square's co-ordinates are (0,0) (1,0) (1,1) and (0,1) ) , ??
Well firstly I found the complementary probability, the probability that its closer to the face instead :/I got:
(4*sqrt(2)-5)/3 and am pretty confident about it...
Yeah your equations are pretty similar to mine. Did mine quite carefully so am pretty sure its right. Will try the 3d one in a sec.Well firstly I found the complementary probability, the probability that its closer to the face instead :/
And even then I still get the wrong answer because I don't know to find the area of a triangle properly and used the wrong base length.
I think it should be, and I'm not bothered to do the calculation again
The above is supposed to be 4 times the area of the region bounded by the cartesian equations:
dem mistakes