Solve Optimization Problem: Minimize Cost of New Highway

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In summary, the conversation discusses a problem where a city needs to build a new section of highway to connect an existing bridge and highway interchange. The cost of building over marshland is $5 million per mile, while over dry land it is $2 million per mile. The optimal point for the highway to emerge from the marsh is found using the Pythagorean theorem and the cost function. The savings from this point compared to building a straight line between the two points is approximately $1.96 million. Part b) of the problem involves reestimating the cost over marshland and finding the new optimal point and savings.
  • #1
thedude23
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Now I hate optimization problems and I cannot figure this one out at all.

1)

A city wants to build a new section of highway to link an
existing bridge with an existing highway interchange, which
lies 8 miles to the east and 10 miles to the south of the bridge.
The first 4 miles south of the bridge is marshland. Assume
that the highway costs 5 million per mile over marsh and 2 million per mile over dry land. The highway will be built
in a straight line from the bridge to the edge of the marsh,
then in a straight line to the existing interchange.

a)

At what point should the highway emerge from the marsh in order to
minimize the total cost of the new highway? How much is
saved over building the new highway in a straight line from the
bridge to the interchange?

b)
Just after construction has begun on the highway, the cost per km over marshland is reestimated
at $6 million per km. Find the point on the marsh/dryland boundary that would
minimize the total cost of the highway with the new cost estimate. How much would be saved
over continuing with the planned route?

My try:
Find the two triangles:

So 8 miles east and 4 miles south for the marsh.

8-x east and 6 miles south or the dryland.

So:
C(x)= (√(8^2-4^2))(5000000) + 2000000(√((8-x)^2-(6)^2))

Im pretty sure I am doing this wrong.
 
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  • #2
I would begin with a diagram:

View attachment 4972

Also, let's let the cost function $C(x)$ be in millions of dollars. In the diagram $M$ is the distance across marshland, and $D$ is the distance across dry land. Using the Pythagorean theorem, we then find:

\(\displaystyle C(x)=5\sqrt{x^2+4^2}+2\sqrt{(8-x)^2+6^2}\)

So, now you want to equate $C'(x)$ to zero and solve for $x$...what do you find?
 

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  • #3
Performing the differentiation and equating the result to zero, we obtain:

\(\displaystyle C'(x)=\frac{5x}{\sqrt{x^2+16}}+\frac{2(x-8)}{\sqrt{(x-8)^2+36)}}=0\)

Observing that for all real $x$, we have:

\(\displaystyle 0<x^2+16\) and \(\displaystyle 0<(x-8)^2+36\)

we may then multiply through by \(\displaystyle \sqrt{\left(x^2+16\right)\left((x-8)^2+36\right)}\) to obtain:

\(\displaystyle 5x\sqrt{(x-8)^2+36}+2(x-8)\sqrt{x^2+16}=0\)

which we may arrange as:

\(\displaystyle 5x\sqrt{(x-8)^2+36}=2(8-x)\sqrt{x^2+16}\)

Square both sides (and keep in mind in doing so we may introduce extraneous solutions):

\(\displaystyle 25x^2\left((x-8)^2+36\right)=4(x-8)^2\left(x^2+16\right)\)

Distribute:

\(\displaystyle 25\left(x(x-8)\right)^2+900x^2=4\left(x(x-8)\right)^2+64(x-8)^2\)

Arrange as:

\(\displaystyle 21\left(x(x-8)\right)^2+900x^2=64(x-8)^2\)

Square binomials:

\(\displaystyle 21x^2\left(x^2-16x+64\right)+900x^2=64\left(x^2-16x+64\right)\)

Distribute:

\(\displaystyle 21x^4-336x^3+1344x^2+900x^2=64x^2-1024x+4096\)

Arrange quartic polynomial in standard form:

\(\displaystyle 21x^4-336x^3+2180x^2+1024x-4096=0\)

Using a numeric root finder (and discarding complex and negative roots), we obtain:

\(\displaystyle x\approx1.25290753523819\)

To ensure this is not an approximation of an extraneous root, we should check it:

\(\displaystyle f(x)\equiv5x\sqrt{(x-8)^2+36}\implies f(1.25290753523819)\approx56.5626548992099\)

\(\displaystyle g(x)\equiv2(8-x)\sqrt{x^2+16}\implies g(1.25290753523819)\approx56.5626548992098\)

This approximation checks out. Now we need to show that this critical value minimizes the cost function. Using the first derivative test, we find:

\(\displaystyle C'(1)=\frac{5}{\sqrt{17}}+\frac{2(-7)}{\sqrt{85}}=\frac{5\sqrt{5}-14}{\sqrt{85}}<0\)

\(\displaystyle C'(2)=\frac{5}{\sqrt{5}}+\frac{2(-6)}{6\sqrt{2)}}=\frac{5\sqrt{2}-2\sqrt{5}}{\sqrt{10}}>0\)

Thus, we know the critical value is at a minimum.

Now to find the savings realized by using this value of $x$ rather than using a direct route between the two points, we first need to find the value of $x$ in this scenario, and we can use similarity to do so:

\(\displaystyle \frac{4}{x}=\frac{6}{8-x}\implies x=\frac{16}{5}\)

And so the savings $S$ (recall this is in millions of dollars) is found by:

\(\displaystyle S\approx C\left(\frac{16}{5}\right)-C(1.25290753523819)\approx1.963789989453947427\)

And so the savings to the nearest penny is: \$1,963,789.99.

I will leave it to the reader to try part b). :D
 

FAQ: Solve Optimization Problem: Minimize Cost of New Highway

How do you define the optimization problem in this scenario?

In this scenario, the optimization problem is defined as finding the most cost-effective solution for building a new highway, while meeting all necessary requirements and regulations.

What factors are considered when trying to minimize the cost of a new highway?

Several factors are considered in minimizing the cost of a new highway, including the length and width of the highway, the type of materials used, the terrain and topography of the area, the necessary infrastructure and accommodations, and any potential environmental impacts.

How do you determine the optimal route for the new highway?

The optimal route for the new highway is determined through a process of analysis and evaluation, taking into account various factors such as distance, traffic patterns, and potential obstacles. This may involve using computer models and simulations to compare different routes and select the most efficient and cost-effective option.

What are some common challenges in minimizing the cost of a new highway?

Some common challenges in minimizing the cost of a new highway include balancing the need for cost savings with safety and quality standards, dealing with unexpected obstacles or complications during construction, and addressing potential environmental concerns and community opposition.

What are the potential benefits of minimizing the cost of a new highway?

Minimizing the cost of a new highway can bring several benefits, such as reducing the financial burden on taxpayers, making the project more financially feasible and attractive to potential investors, and potentially freeing up funds for other infrastructure projects. It can also help to minimize the environmental impact of the project and make it more socially and politically acceptable to the community.

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