Please do not copy from previously answered questions.

You can ignore the modelling aspect of the questions. I just need help with the mathematical aspect. Thanks

[2] Question 6 28 marks The aim of this question is to assess your ability to create and solve a mathematical model. Two towns, A and B, are 50 km apart and have populations of 5000 and 135 000, respectively. Lyra is an amateur astronomer who wishes to set up her telescope at a position on a straight road between A and B such that the total light pollution resulting from the towns is a minimum. (a) Specify the purpose of a mathematical model for Lyra's problem. (b) Assuming that the light pollution in the sky caused by the lights in a town is directly proportional to the population of the town and inversely proportional to the square of the distance from the town, create a mathematical model for Lyra's problem. State any assumptions that you make, and define any variables that you use. Solve the problem, and interpret your solution. Will, another amateur astronomer, wishes to set up his telescope between four towns, P, Q, R and S, at a point where the light pollution is a minimum. These towns are situated at the four corners of a rectangle, and their positions in the (21, 22)-plane are given by P(0,0), Q(0,14), R(19,14) and S(19,0), where the units of the coordinates are kilometres. The populations of P, Q, R and S are 4000, 3000, 1000 and 1000, respectively. (c) Revise your model from parts (a) and (b) to take account of Will's situation. Use an appropriate Maxima worksheet with a grid search to show that the total light pollution from the towns is at a minimum at approximately (13,7). You may assume that there is a single stationary point, which corresponds to a global minimum, within the rectangular region with vertices at (3,3), (3,11), (16,11) and (16,3), and you should restrict your grid search to this region. Hence determine the position where Will should set up his telescope, to an accuracy of two decimal places. [15] 5