# Calculus 1 Courses's Optimization of Pipeline Route Problem, Produces Strangely Complicated Value.

I am a Calculus 1 student and I have an optimization word-problem that is giving me a lot of trouble.

It has two variables. I have found the value for $$y$$, but when I plugged it into the equation and tried to solve for the $$x$$ I couldn't find it's value. I used Symbolab to solve it, but it came up with a decimal number that's extremely complicated when written as a fraction. My professor has given us very complicated problems before, but the complexity of this number is such that I feel like it's very likely I did something wrong.

I have checked other parts of my work with Symbolab and I am still not sure where I went wrong, but I would really appreciate it if you would take a look and determine if there are any parts that don't look right to you.

An oil refinery is located on the north bank of a straight river which is $$2$$km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river $$6$$km east of the refinery. The cost of laying pipe is $$\400,000$$ per km over land to a point $$P$$ on the north bank and $$\800,000$$ per km under the river to the tanks. To minimize the cost of the pipeline, where should $$P$$ be located?

$$P=$$ The area where the pipeline enters the river.

$$x=$$ The horizontal distance between the oil refinery and the storage tanks.

$$y=$$ The euclidean distance between $$P$$ and the storage tanks.

The Pythagorean theorem states $$2^2+(6-x)^2=y^2$$.

The cost of the pipeline is $$C = 400,000x+800,000y$$.

finding $$y$$:

$$4+(6-x)^2 = y^2 \to y= \pm \sqrt{4+(6-x)^2}$$

Differentiating $$C = 400,000x+800,000\sqrt{4+(6-x)^2}$$:

$$\frac{d}{dx}\sqrt{4+(6-x)^2}=\frac{1}{2\sqrt{4+(6-x)^2}}\cdot-2(6-x)=\frac{-(6-x)}{\sqrt{4+(6-x)^2}}$$

$$\frac{d}{dx}800,000\sqrt{4+(6-x)^2}=[\frac{-(6-x)}{\sqrt{4+(6-x)^2}}\cdot800,000] = \frac{-800,000(6-x)}{\sqrt{4+(6-x)^2}}$$

$$\frac{d}{dx}400,000x+800,000\sqrt{4+(6-x)^2}=400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}}$$

Setting $$400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 0$$ and solving for $$x$$.

$$x=4.84530..$$

I'm not completely sure how to write the fraction out with math notation here because a single square root seems to cover part of the numerator and all of the denominator, but you can see it if you plug $$400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 0$$ into Symbolab's "solve for" calculator.

• You equations are correct. The answer is indeed $6 - \frac2{\sqrt{3}} \approx 4.8453$ and there's nothing complicated about it. – Lee David Chung Lin Apr 12 at 22:49
• @LeeDavidChungLin Thank you! Symbolab came up with something really weird so I got freaked out. – LuminousNutria Apr 12 at 22:51

As a commenter has made clear, when solving $$400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 0$$ for $$x$$, $$x=6-\frac{2}{\sqrt{3}}$$.

Steps Involved:

$$400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 0$$

Subtract $$400,000$$ from both sides.

$$\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 400,000$$

Divide both sides by $$400,000$$.

$$\frac{2(6-x)}{\sqrt{4+(6-x)^2}}=1$$

Multiply both sides by $$\sqrt{4+(6-x)^2}$$.

$$2(6-x)=\sqrt{4+(6-x)^2}$$

Take both sides to the power of $$2$$

$$4(6-x)^2=4+(6-x)^2$$

Subtract $$(6-x)^2$$ from both sides.

$$3(6-x)^2=4$$

Divide both sides by $$3$$.

$$(6-x)^2=\frac{4}{3}$$

Square both sides.

$$6-x=\frac{2}{\sqrt{3}}$$

Subtract $$6$$ from both sides.

$$x=6-\frac{2}{\sqrt{3}}$$

• You lost the $\sqrt 3$ – Claude Leibovici Apr 13 at 5:58
• @ClaudeLeibovici Whoops, I have fixed it. – LuminousNutria Apr 13 at 17:30