Pipe Needed for Natural Gas LineDate: 09/03/97 at 16:46:19 From: George Santana Subject: Measuring distances Two houses are 130 metres apart at distances of 55 and 105 metres for a straight natural gas line. The gas company decides to service both houses by tapping into the main line at only one place. The minimal amount of pipe needed to service both houses (to the nearest metre) is A 200 B 215 C 237 D 202 E 198. Date: 09/03/97 at 18:49:34 From: Doctor Anthony Subject: Re: Measuring distances Draw the figure and note that a line joining the house at 105 metres from the gas pipe to the tapping point, if extended, should pass through a point 55 metres from the gas pipe but on the opposite side from the second house. If you reflect the second part of this line in the line of the gas pipe you end up at the second house. It is easy to see that this will be the shortest possible line since is initially a straight line (before reflecting), and it is the total distance of the piping required to connect the two houses. A little simple geometry shows that the two points on the main gas line closest to the two houses are 120 metres apart. You can make a right angled triangle with sides 130 and 50 metres (difference between 105 and 55), so that third side is 120. We have, in effect, a 5, 12, 13 triangle. If the tapping point is x metres along the main gas pipe from the closest point to the 55-metre house, and y is the distance along the main gas pipe to the closest point of the 105-metre house, then similar triangles shows that x/y = 55/105 so y = (105/55)x also x + y = 120 x + 105/55 x = 120 55x + 105x = 6600 160 x = 6600 x = 41.25 then y = 78.75 Using Pythagoras we can now find the total length of piping connecting the tapping point to the two houses. From the 55-metre house, pipe distance = sqrt(41.25^2 + 55^2) = 68.75 From the 105-metre house, pipe distance = sqrt(78.75^2 + 105^2) = 131.25 Total pipe length = 68.75 + 131.25 = 200 metres. This is answer A. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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