Answer Happy

You are tasked with designing an open-top trough to hold the most volume of liquid, subject to an area constraint. This is most efficiently achieved by bending a rectangular metal sheet of length L and width W by some angle o along the center line shown shown in the figure below. The supplier of the metal sheet charges per square meter, and so the ends of the trough must be included. The overall area of metal obtained is 10 m2. W L (a) The function and constraint which model the problem take the general forms: = d?b sin e - a² V= 4 1 10 m2 = ab + sin e 2 Draw a clearly labelled diagram of the cross section of the trough and derive the definitions of a and b in terms of W and L [5 marks] (b) By discussing level curves of a function to be optimised f and the constraint 8 = 0, explain how we arrive at the Lagrange method of constrained optimisation. Define any relevant terms. You may wish to include any sketches to clarify. (Max. 100 words) [5 marks] (c) Using your answer from part (a), show how this can be represented in a form which can be reduced to a system of equations, through the Lagrange method of constrained optimisation (e.g. Vf = 1Vg). If you were unable to obtain a solution to part (a), use the form given in part (a). [6 marks] (d) Considering each equation from part (c) and the constraint, solve the system of equations and determine the dimensions of the trough which maximise the volume. Ensure that you report only physically realisable dimensions. [5 marks] (e) Over time, a thin film of algae begins to form over a corner portion of the surface of the liquid contained in the trough. The surface density of this film, in units of gm-2, is given by: 1 p(x, y) = 1 + y3' and the region over which the algae forms is given by: 0 < x < 4m and Vx sy s 2 m. Sketch the region of integration and determine the mass of algae formed. NB: changing the order of integration may make the integral easier to evaluate. [9 marks]