A school marks out an L-shaped running lane on the field for sports day using field paint. The diagram shows the boundary of the lane, with four of its six ...
Question 1 Report
A school marks out an L-shaped running lane on the field for sports day using field paint. The diagram shows the boundary of the lane, with four of its six sides labelled in metres, where \(x\) is a positive number.
- Find expressions, in terms of \(x\), for the lengths of the two unlabelled sides. (2)
- Hence show that the perimeter of the lane is \(8x + 16\) metres. (2)
Answer Details
The running lane is an L-shaped hexagon: the two horizontal sides at the top must together span the same total width as the bottom, and the two vertical sides on the left must together span the same total height as the tallest side.
The full width of the lane is the bottom side, \(3x+1\), and the top step is \(x+4\), so the remaining unlabelled horizontal side is \[(3x+1)-(x+4)=2x-3 \text{ metres}\] The full height of the lane is the left side, \(x+7\), and one internal vertical step is \(2x-2\), so the remaining unlabelled vertical side is \[(x+7)-(2x-2)=9-x \text{ metres}\] [2 marks]
Adding all six sides of the boundary gives the perimeter: \[P=(3x+1)+(x+7)+(x+4)+(2x-2)+(2x-3)+(9-x)\] Collecting the \(x\) terms, \(3x+x+x+2x+2x-x=8x\), and collecting the number terms, \(1+7+4-2-3+9=16\), gives \[P=8x+16 \text{ metres}\] as required. [2 marks]
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