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. (b) The architect then considers the shape where the semi-circle doesn't reach the full width of the rectangle, though is still centred. Set up this problem that is similar to that of part (a) but with an extra variable. The want to maximise the area becomes more subtle as with the option to vary the size of the semi-circle now included, this means could have a semi-circle of radius 0. In other words, the option to not remove any semi-circle at all is included in the problem, and this would indeed maximise the area of the window. The architect still decides to do some investigating with the area S and the imposition of the xed contour only. Here the architect nds that there is a local maximum to maybe consider. Find this local maximum area along with associated measurements, prove its nature and discuss its feasibility in terms of the other constraints. Working may be done with exact values or with numbers correct to four decimal places. Any nal measurements should be given correct to two decimal places