A hemispherical tank of radius $$2$$ metres is initially full of water and has an outlet of $$12$$ cm² cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $$v(t)=0.6$$ $$\sqrt {2gh\left( t \right),} $$ where $$v(t)$$ and $$h(t)$$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time $$t,$$ and $$g$$ is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: From a differential equation by relasing the decreases of water level to the outflow).

Let $$b \ne 0$$ and for $$j=0, 1, 2, ..., n,$$ let $${S_j}$$ be the area of
the region bounded by the $$y$$-axis and the curve $$x{e^{ay}} = \sin $$ by,
$${{jr} \over b} \le y \le {{\left( {j + 1} \right)\pi } \over b}.$$ Show that $${S_0},{S_1},{S_2},\,....,\,{S_n}$$ are in
geometric progression. Also, find their sum for $$a=-1$$ and $$b = \pi .$$

Evaluate $$\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.$$

If $$\Delta $$ is the area of a triangle with side lengths $$a, b, c, $$ then show that $$\Delta \le {1 \over 4}\sqrt {\left( {a + b + c} \right)abc} $$. Also show that the equality occurs in the above inequality if and only if $$a=b=c$$.