Math Investigation: The Lawn Two men Tom and Dom were going to cut the grass on their square lawn.

1. Border 3m Math Investigation: The Lawn • Two men Tom and Dom were going to cut the grass on their square lawn. They agreed that each should cut one-half of the area. Tom went first and cut a border 3 meters wide all the way round as shown in the diagram. After doing a few calculations, Dom agreed that exactly one-half had been cut and happily cut the rest. What was the area of the lawn?

2. Solution:- The Lawn Let x = length of the outside square. Then length of inside square is x-6. So the area of bit cut = Area inside square x2 - (x-6)2 = (x-6)2 Rearranging gives x2 – 24x +72 = 0 This requires the quadratic formula for its solution. This gives x = 20.49 and the corresponding area as 419.8m2

3. Exercises Exercise 1. Suppose the garden was such that the border cut by Tom had actually been 5 metres wide all the way round, what would the area of the lawn be now? Exercise 2. If the border cut by Tom had been ‘a’ metres all the way round, show that the area of the lawn is given by the expression A = 8a2(3+2√2).