Pipe Needed for Natural Gas Line

Date: 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
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