A gas pipe line is to be constructed from a storage tank, which is right on a road, to a house which is
600 feet down the road from the tank and 300 feet set back from the road. Pipe laid along the road cost
$8.00/ft while the pipe laid off the road costs $10.00/ft. What is the minimum cost for which this pope
line can be built? Make an objective function with two variables and state the constraints
So, draw the diagram. If the pipeline is in straight segments, and leaves the road x feet from the tank, then the overland distance is
√((600-x)^2 + 300^2)
so the cost of the pipeline is
c(x) = 8x + 10√((600-x)^2 + 300^2)
dc/dx = 8 - 10(600-x)/√((600-x)^2 + 300^2)
dc/dx=0 at x=200, so the minimum cost is c(200) = $6600
how did you get x =200??
To find the minimum cost for constructing the gas pipeline, let's define the following variables:
Let x = length of pipe laid along the road (in feet).
Let y = length of pipe laid off the road (in feet).
Objective function:
Cost = 8x + 10y
Constraints:
1. Total length of the pipeline: x + y = 600 feet
2. Distance of the house from the road: y = 300 feet
Now we can solve the problem by substituting the second constraint into the first constraint.
To find the minimum cost for which the gas pipeline can be built, we need to create an objective function with two variables and state the constraints. Let's use the following variables:
Let x represent the length of the pipe laid along the road (in feet).
Let y represent the length of the pipe laid off the road (in feet).
Objective Function:
The objective is to minimize the cost of the pipeline, which is determined by the length of pipe laid along the road and the length of pipe laid off the road. Therefore, the objective function is:
Cost = 8x + 10y
Constraints:
1. The total length of the pipe laid along the road and off the road should be equal to the distance between the storage tank and the house. Therefore, the first constraint is:
x + y = 600
2. The house is set back 300 feet from the road, so the length of the pipe laid off the road (y) should be 300 feet less than the length laid along the road (x). Thus, the second constraint is:
x - y = 300
To summarize, we have the following objective function with two variables and two constraints:
Objective Function: Cost = 8x + 10y
Constraints:
1. x + y = 600
2. x - y = 300