The diagram shows the cross-section of a high-tech "Aero-Shelter" designed for Antarctic research. The shelter sits on a flat ice-sheet which acts as a horizontal base.
The cross-section of the shelter begins at the origin O O\,O and ends at point P P\,P on the xxx-axis. The shape is modeled by the equation
x2+2y2=kx+10y x^2 + 2y^2 = k\sqrt{x} + 10y x2+2y2=kx+10ywhere k k\,k is a constant, and x x\,x and y y\,y are the horizontal and vertical distances in metres from the origin OOO.
Given that the total width of the shelter base OP OP\,OP is 16 metres, determine the value of the constant kkk.
Show that the maximum height of the Aero-Shelter above the ice-sheet is approximately 10.7 metres.
State one limitation of using this mathematical equation to model the physical structure of the shelter.
