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### 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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Word Problems

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