Consider an ant crawling along the curve $$\,{\left( {x - 2} \right)^2} + {y^2} = 4,$$ where $$x$$ and $$y$$ are in meters. The ant starts at the point $$(4, 0)$$ and moves counter $$-$$clockwise with a speed of $$1.57$$ meters per second. The time taken by the ant to reach the point $$(2, 2)$$ is _________ (in seconds).

Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, = \,\,\sin t,\,\,\,z\left( t \right)\,\, = \,\,{2 \over \pi }t,\,\,\,0 \le t \le {\pi \over 2}.$$ The length of the curve is ________.

The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is

A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation $$y = 2x\,\,\, - 0.1{x^2}$$ where $$y$$ is the height of the arch in meters. The maximum possible height of the arch is