A home in rural area located at A (22,-24) IS CONVERTING TO NATURAL GAS HEATING. They will be connecting the home to a main gas line with the end points G (-16,-30) and H (38,42) Each unit represents 1m. At what point should the builder connect to the main gas lines to minimize cost.
Find the general equation of the line through G and H, using
your favourite method.
You want the shortest distance from point A to the line you found
If the coordinates of a point are (a,b)
then the shortest distance to Ax + By + C = 0
= |aA + bB + C|/√(A^2 + B^2)
once you have the equation, just plug in and grind it out
Alright I found the answer I believe. Thanks
To find the point where the home should be connected to the main gas line to minimize cost, we need to calculate the distances between the home location (point A) and the two end points of the main gas line (points G and H).
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances between point A and points G and H:
For point G:
d_AG = sqrt((-16 - 22)^2 + (-30 - (-24))^2) = sqrt((-38)^2 + (-6)^2) = sqrt(1444 + 36) = sqrt(1480) = 38.48
For point H:
d_AH = sqrt((38 - 22)^2 + (42 - (-24))^2) = sqrt((16)^2 + (66)^2) = sqrt(256 + 4356) = sqrt(4612) = 67.92
Now, we compare the distances to determine the closer point:
If d_AG < d_AH, then point G is closer and the home should be connected to point G.
If d_AH < d_AG, then point H is closer and the home should be connected to point H.
In this case, d_AG < d_AH, so the home should be connected to the main gas line at point G (-16, -30) to minimize cost.