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+800,000y$ :

$ $ \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.