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August 12th, 2014, 04:03 PM  #1 
Senior Member Joined: Jan 2012 Posts: 724 Thanks: 7  Calculate the length of the field.
A rectangular playing field 18m wide. It is surrounded by a path 6m wide such that its area is equal to the area of the path. Calculate the length of the field. Width of path and field = 18m + 12m = 30m $\text{Area of one path} \ = 30\times6=180m^2\\\text{Area of second path}=180m^2\\\text{Total area of the two path}=360m^2$ We are told that the area of field is equal to area of path. So how do I get the area of the path and of field respectively so that I can equate them and from there get the lenght of the field? 
August 12th, 2014, 04:17 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,559 Thanks: 2561 Math Focus: Mainly analysis and algebra 
I think there is path on all four sides of the field. Set the length of the field to be x Create expressions for the area of the field and the area of the field + path. Use those expressions, together with what you are told about the area of the path to get an equation for x. I get x a little greater than three times the width. Last edited by v8archie; August 12th, 2014 at 04:20 PM. 
August 12th, 2014, 08:35 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,799 Thanks: 970 
p = path width (6) w = width (18) x = length (?) area field = area path wx = 2p(x + 2p) + 2pw ; simplify: x = 2p(2p + w) / (w  2p) Finish it... 
August 14th, 2014, 06:47 PM  #4 
Senior Member Joined: Jan 2012 Posts: 724 Thanks: 7 
$\text{Area of one path} \ = 30\times6=180m^2\\\text{Area of second path}=180m^2\\\text{Total area of the two path}=360m^2$ Area of the other two paths = $12Lm^2$ Area of field = $18Lm^2$ $\therefore18L=12L+360\\\longrightarrow\\L=60m$ 

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calculate, field, length 
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