please help with answering these questions, thank you!
Khanh has a property and owns several horses, that are housed in a stable (dimensions are 12m x 20m). To provide them more space Khanh wishes to build a yard that the horses will be free to roam in. The diagram below illustrates the existing stable and the proposed fence that will enclose the yard. Khanh has several "requirements for the yard", that must be obeyed: Khanh has access to 200 metres of fencing. This fencing required for both the yard and a 'breaking in' enclosure (details for this provided below). The fence enclosing the yard must extend out from opposite corners of the stable at right angles to the stable wall (see diagram above). Khanh also requires a smaller 'breaking in' enclosure, that must be completely contained within the yard. However, it can share a boundary with the yard or stable (for instance, it can be built next to the stable to reduce the required amount of fencing). The size of this enclosure must be at least 2252. The total area of the yard must be as large as possible, whilst meeting the above requirements. For questions 1 to 6, we are going to assume that the 'breaking in' enclosure is a square area that doesn't share any boundaries with the yard or stable. If the length of the fence extending out at the North southern end of the yard is x metres. Yard The diagram below illustrates the scenario with the Breaking In assumed square 'breaking in' enclosure. enclosure (A = 225m) 12m Stable 20m X Questions: (19 marks total) 1) Determine the lengths of all the sides of the yard in terms of x (remember some fencing needs to be used to make the 'breaking in' enclosure). 2) Draw a new diagram including the 'breaking in' enclosure and label all the sides with appropriate lengths. 3) Write an expression for the area of the yard, A(x), in terms of x. (2) 4) Considering the scenario, what values can x be? (ie. What is the restricted domain for A(x)?) (1) 5) Using technology draw the graph of A(x), for the domain identified in (3). (2) 6) Using calculus, determine the value of x that will provide the largest possible area for the yard. (2) 7) Provide a final diagram including all relevant details and dimensions. (2) Extension It is possible for the 'breaking in' enclosure to have a different location and dimensions, whilst maintaining the required area of 225m. Determine the optimal dimensions of the yard, that will provide the largest area, whilst still meeting all the requirements listed earlier. (5) Note - Considering the mark allocation for this question. I recommend you consider at most three other options and choose from those (be strategic in which options you do calculations for)
