Power Plant on the River Problem
As the construction manager for the Lightem Up Power Company , you make decisions to hep minimize costs of the company's construction projects. Today you've been asked to determine the path of a pipe that will house communication lines from the factory to the power plant pictured here.
Factory . F_______7 miles_______
| river |
4 miles
|______________________|
.P Power
Plant
You must connect a pipe from point F at the factory to point P at the power plant. It will cost $12 per foot to lay the pipe on land and $21 per foot to lay the pipe in the water. Where should the pipe be placed so as to minimize the total cost of laying the pipe?
let the point directly across from P (the power plant) be Q.
then FQ = 7 miles
let R be a point between F and Q so that RQ = x miles, and FR = 7-x miles.
RP^2 = x^2 + 16
RP = (x^2+16)^(1/2)
So the path of the pipeline is PR + RF.
Cost of pipe on land: $12 per foot = $12(5280) per mile
= $63360 per mile
Cost of pipe in water: $21 per foot or
$110880 per mile
Cost = 110880*RP + 63360*FR
= 110880(x^2+16)^)1/2) + 63360(7-x)
d(Cost)/dx = (1/2)(2x)(110880)(x^2+16)^(-1/2) - 63360x
= 0 for a minimus of Cost
this reduced to
7x/√(x^2+16) = 4 (I divided by 15840x)
cross-multiplying and squaring both sides gave me
49x^2 = 16x^2 + 256
33x^2 = 256
x = 16/√33 = 2.785
then 7-x = 4.215
They should aim for a point 4.215 miles from the factory, then cross the river to the power plant.
To determine the path of the pipe that will minimize the total cost of laying the pipe, we need to compare the cost of laying the pipe on land to the cost of laying the pipe in the water.
Let's start by calculating the cost of laying the pipe on land. Given that the distance from the factory (F) to the river is 7 miles, and the distance from the river to the power plant (P) is 4 miles, the total distance of the land portion of the pipe is 7 + 4 = 11 miles.
Since the cost of laying the pipe on land is $12 per foot, we need to convert the distance from miles to feet by multiplying it by 5280 (1 mile = 5280 feet). Therefore, the total cost of laying the pipe on land is 11 * 5280 * $12 = $696,960.
Now let's calculate the cost of laying the pipe in the water. The length of the river portion of the pipe is the same as the length of the river, which is 4 miles. Since the cost of laying the pipe in the water is $21 per foot, we multiply the distance in miles by 5280 to convert it to feet. Thus, the total cost of laying the pipe in the water is 4 * 5280 * $21 = $443,520.
To determine where the pipe should be placed to minimize the total cost, we compare the costs of laying the pipe on land and in the water. In this case, the cost of laying the pipe in the water is lower than the cost of laying it on land. Therefore, the pipe should be placed in the water, following the path of the river, to minimize the total cost of laying the pipe.
Note: The problem assumes that the cost per foot remains constant for both land and water portions of the pipe. In real-life scenarios, there could be other factors to consider, such as terrain conditions or engineering requirements, which may affect the cost of laying the pipe.