Mower Review

- Mathematics puzzle...give a full description for a full rating....?
You have been hired by an organisation with a large lawn, a robotic lawn mower and a part-time gardener. They want you to determine a strategy for minimising the total cost of mowing the entire lawn each Saturday based on the following constraints: (a) The lawn mower is cheap to run but it can only move in straight lines over the lawn and reflects perfectly each time it crosses a boundary. However you can choose its initial orientation when it is first switched on as well as the total number of passes it will make over the lawn. (b) The part-time gardener is expensive to hire but can cut grass anywhere in the lawn. (c) The entire lawn must be mowed by the end of the day, so the lawn mower can make no more then fourty passes across the lawn. You have been given a simulator which accurately models the behaviour of the problem, which can be accessed here: http://www.cs.adelaide.edu.au/~josephk/PBL/AutoMower.html Experiment with the simulator to determine the best choices for the initial entry angle and the total number of passes the mower should make. Justify your answer, explaining how you determined your values were the best choices available.

A More information is required to solve this puzzle, such as width of mower, size and shape of the garden, speed of the mower, speed of the gardener and the costs for the mower and the gardener. Let's assume the lawn to be rectangular with width w and length l, the initial orientation to be at an angle a to the side of the lawn and the width of the mower to be m. After two passes (one to and one fro), the mower will be 2*w*tan a. If this distance is equal to the width of the mower, the mower will cut through all the grass on the lawn (assuming it gets sufficient passes) except a small traingular area at the top. If we could get the gardener to just reposition the mower at the end of either the first or the last pass, it will cut that area as well and since the gardener is more expensive, proabably it will be a cheaper choice. Now, 2*w*tan a = m or tan a = m/2w. If m is very small compared to w (which it should be) and a is in radians, the above can be approximated to a=m/2w. Number of passes required = l/w or l/w+1 if the gardener repositions it. In the simulator, 40 passes are much more than enough for mowing.