Hi, I'm currently in Pre-Calculus in High school, a course prepping for university; I'm hoping this thread is in the appropriate section! I apologize for the long question, but I don't want to confuse anyone who is willing to help me.

We are currently doing a project on minimizing the cost for a company that has been contracted to lay pipeline through a forested area between a natural gas well and a storage facility.

Picture a rectangle where the horizontal is the roadway, the vertical is always 3000 m long, and in the middle is where a forest is. There's a diagonal pipeline starting from the lower left corner of the rectangle (the well site) to the upper right corner of the rectangle (the storage facility). It wil cost $230/m to lay pipeline parallel to the roadway (horizontal line) and $260/m to lay pipeline through the forested area (diagonal line).

However the diagonal line isn't directly connected from the well site to the storage facility--the company would lay 500 m of pipeline parallel to the roadway while the rest of the pipeline through the forested area (diagonal line). The horizontal line begins at 10 000 m, but decreases every 500 m as the pipeline parallel to the horizontal increases every 500 m.

First I designed a spreadsheet with values. The first column (a) is 'trial number' [let's call this y], the second (b) is 'distance of pipeline parallel to roadway in meters' [let's call this x], third is 'cost of pipeline through forested area in $' (c), fourth (d) is 'cost of pipeline through forested area in $' fifth (e) is 'cost of pipeline through forested area in $' and last (f) is 'total cost of pipeline in $'. For example one of the rows would be:

1 // 500y // √[((10 000 - x)²) + (3000²)] // {√[((10 000 - x)²) + (3000²)]} • 260 // x • 230 // [{√[((10 000 - x)²) + (3000²)]} • 260] + [x • 230]

Simply put it would look like:

1 // 500a // √[((10 000 - x)²) + (3000²)] // 260 • c // 230 • x // d + e

I have to do this for about 25 trials until the distance of pipeline parallel to roadway is 10 000 m. I found that the least expensive path was when the pipeline parallel to the roadway is 4 500 m.

I have to find the formula first for the equation using technology (I haven't gotten that far yet but I know what to do), but once I find the formula my next task is to find the least expensive path using calculus and algebra.

My question is: I'm not exactly sure how to set this question up. Basically I have a rectangle where the vertical is always 3000 m. I have a diagonal line attached to another horizontal line; the horizontal line increasing 500 m while the horizontal line of the rectangle decreases 500 m. I know my solution would probably have to end up being 4 500.

I'm not asking for a full solution (if you're not comfortable with that), but I would be very grateful for any help or tips on solving this.

Thank you very much and again, I'm sorry that this question is long!