Question

The density of a thin circular plate of radius 2 is given by p(x,y)=4+xy.The edge of the plate is described by the parametric equations x=2cost,y=2sint for t in [0,2pi].At what point(s) on the edge of the plate is the density a maximum?

Answer 1

\[dp/dt=-4\sin^2t+4\cos^2t\] Do I set dp/dt=0? And if I do is the correct answer t=pi/4? so that yields the point \[(\sqrt{2},\sqrt{2})\] as the only point on the edge of the plate where the density (p(x,y)) is a maximum?

Answer 2

And is \[\sin^2t=\cos^2t\] the same as
\[\sqrt{(sint)^2}=\sqrt{(cost)^2}\]===>sint=cost. So t=pi/4 the only answer. Is this correct?

Answer 3

1º part looks correct
2º not so sure.....

Answer 4

It should be\[\sin{t}=\pm\cos{t}\]
Also, CHECK whether they really produce the maximum value by plugging the x and y in p(x,y).

Answer 5

Note:
\[0\leq t< 2\pi\]
so you have to make sure that you exhaust this interval for solutions.