1. An architect is designing a special window for a museum. It consists of a rectangle of base .7: and height h where a semi-circle is then removed from the top. A particular building material means the total contour of this exotically shaped window needs to be xed at 600 cm. The architect wants to maximise the area, S', of the window so as to maximise light coming into the room. (a) First, the architect considers the shape where the semi-circle spans the full width of the rectangle. (1) Give an initial set of equations that represent this problem as one equation for the area in terms of a: and it along with four boundary conditions. One of the conditions of this problem is that the area has to be greater or equal to zero. Use only exact values. (ii) The four boundary conditions from part (a)(i) may be condensed to two in terms of :r and h only. Give this condensed set. Then, present the problem again as the equation for the area (in terms of a: and h) along with two boundary conditions (in terms of a: and h only). Use only exact values. (iii) The problem may be written in terms of S and as only. Present this version (using only exact values) and hence nd the measurements a: and h that maximise the area of the window correct to two decimal places. (iv) Consider and solve the problem as originally presented in part (a)(ii) as one requiring the maximisation of a function of two variables :1: and h. You may refer to calculations from previous parts and do not have to redo calculations. (v) Do your answers from part (a)(iii) and part (a)(iv) reconcile? A simple \"yes\" or \"no\" suices