Minimum cost of power line

The derivative of 7500sqrt(x^2+9) is 7500(x^2+9)^-.5(2x), not .5(x^2+9)^-.5(2x)(7500). In summary, the conversation is about finding the minimum cost of building a power line from a town to a mine, given the cost of building along a road and through a forest. The solution involves differentiating a cost equation and solving for x.
  • #1
Glissando
34
0
Hey guys I know this questions has been posted before but I still can't seem to figure it out. Thanks a lot for your help (:

Homework Statement



A power line is to be constructed from a town to a mine that is located near a straight road going to the town. The point on the road closest to the mine is 7.5km from the town and 3km from the mine. The cost of building along the road is 4500$ per km, and through the forest to the mine, 7500$ per km. Find the minimum cost of the power line.


Homework Equations



Cost = Distance of road * 4500 + Distance through forest *7500

The Attempt at a Solution



C = Road(4500) + Forest (7500)
C = (7.5-x)4500 + 7500sqrt(x^2+9)
dC/dx = -4500x + .5(x^2+9)^-.5(2x)(7500)
dC/dx = -4500x + 7500x(x^2+9)^-.5
dC/dx = 500x [-9 + 15(x^2+9)^-.5]

0 = 500x[-9 + 15(x^2+9)^-.5]
0 = -9 + 15(x^2+9)^-.5
9/15 = (x^2+9)^-.5
x = -0.36

Thanks in advance (:
 
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  • #2
Glissando said:
C = (7.5-x)4500 + 7500sqrt(x^2+9)
dC/dx = -4500x + .5(x^2+9)^-.5(2x)(7500)

You have not differentiated correctly.

The derivative of (7.5 - x)4500 is -4500, not -4500x.
 

1. What is the minimum cost of building a power line?

The minimum cost of building a power line can vary depending on various factors such as the length of the power line, terrain, materials used, and labor costs. However, typically the minimum cost can range from $10,000 to $30,000 per mile.

2. How is the minimum cost of a power line calculated?

The minimum cost of a power line is calculated by taking into account factors such as the distance between the power source and the destination, the type of terrain the power line will be built on, the cost of materials and labor, and any additional expenses such as permits and right-of-way acquisition.

3. Can the minimum cost of a power line be reduced?

Yes, there are ways to reduce the minimum cost of a power line. This can be achieved by using cost-effective materials, optimizing the route to minimize distance, and utilizing advanced construction techniques. Additionally, negotiating with landowners for right-of-way access can also help reduce costs.

4. How long does it take to recoup the minimum cost of building a power line?

The time it takes to recoup the minimum cost of building a power line can vary depending on factors such as the cost of electricity in the area, the demand for electricity, and the length of the power line. On average, it can take anywhere from 5 to 15 years to recoup the cost.

5. Are there any environmental concerns associated with building a power line?

Yes, building a power line can have environmental impacts such as deforestation for clearing a pathway, disturbance of wildlife habitats, and potential pollution from construction activities. It is important for power line construction to follow environmental regulations and mitigate any potential negative impacts on the environment.

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