TAKE-HOME PROBLEMS FOR EXAM III

TAKE-HOME PROBLEMS FOR EXAM III

These problems may be done in advance at home and must be turned in at the end of the exam on exam day.

1. For the data points (1,1), (2,3), (3,6), (4,7), and (5,9) find the best fit linear function (y = ax + b) based on a least squares criteria. Show the system of equations to be solved in finding a and b by setting S_a and S_b each equal to zero. S(a,b) is the function giving the sum of the squares of the errors. Make sure you follow the instructions for this problem given in class.

2. Find the box of largest volume that can be inscribed in the ellipsoid

3. In the problem pictured below, determine the values for x and y that will minimize total construction costs. The idea is to lay pipe from point P to Point Q. It costs 3 million dollars per mile to lay the pipe through the blue area, 2 million dollars per mile to lay the pipe through the green area, and 1 million dollars per mile to lay the pipe along the boundary between the green area and the brown area. Consider the colored regions to be rectangles, x to represent the horizontal distance for the pipe in the blue region and y to be the horizontal distance for the pipe in the green region. The blue region and the green region are each 1 mile wide and the horizontal distance from P to Q is 5 miles. You must thoroughly investigate the costs on the boundary of the region over which you would be applying the cost function. You may assume you would not minimize total cost by laying pipe in the negative x direction, negative y direction, or by laying pipe past point Q and then coming back. Thus this region (see the bottom picture on the right) would be described by

Click on the top picture at the right to see an animation of some of the possible paths.

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Lane Vosbury, Math Chair, Seminole Community College email: vosburyl@scc-fl.edu