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This is my first question here. Excuse my extremely informal maths lingo. I don't think I'll be able to classify this properly, but I'm going to say it could be related to calculus. Anyway, the picture that goes with this question (and is needed to find the answer) is here.

The question is as follows:

Dan has purchased a country property with layout and dimensions as shown in the diagram.

$a).$ Show that the property has a total area of $987.5 ha$ (hectares). (which I have done)

$b).$ Dan wants to split the property in half (in terms of area) by building a straight‐lined fence running either north–south or east–west through the property. Assuming the cost of the fencing is a fixed amount per linear metre, justify where the fence should be built (that is, how many metres from the top left‐hand corner and in which direction), to minimise the cost.

It's question b) that is confusing me. I've tried relating it to finding minimum and maximum area, but the shape isn't a rectangle and is a composite figure instead. I am quite stuck on this. Any help is appreciated greatly!

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    Hmm. Possibly investigate a "median" of sorts through the shape? If you could find the equation of said median's length, it would be easy to optimize with Calc. – 2017-01-27
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    @Vedvart1 Ah, I think I've appeared to bite off more than I can chew. Do you mind explaining what you mean by this? Thanks! – 2017-01-27
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    I think i was on the wrong track there, but I'm working on an interesting approach now. I'll post as an answer if it fleshes out to anything. – 2017-01-27
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    @Vedvart1 Okay! I look forward to seeing your approach, thanks for the help! – 2017-01-27

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Okay, not sure if this is perfect, as we've assumed some things. I'm gonna use my "diagram" here to explain what I'm doing (sorry for the MS Paint lol). Let's draw that green line to divide the area into two. Since each half is equal in area, I'd say it's pretty safe to assume that the green line should be in the bigger section (the square) and either $x > 1000$ or $y >1000$ (if neither were, the green line would dip into that handle there; if one is, we can just flip x and y, area staying the same.) Now we calculate all the areas. For the biggest section, I calculate the area of the square minus the triangle we cut off and the corner already cut off. We know the two halves are equal, so $$\frac{xy}{2} + 2,000,000 = 9,000,000 - \frac{xy}{2} - 1,125,000$$ $$xy = 5,875,000$$ We want to minimize the length of the green line, $\sqrt{x^2+y^2}$. From here it's just a simple Calc optimization problem.

EDIT: I don't think we need to assume that the green line can't be in the handle. Even with the green line running from the top left corner to the inner corner where the handle and square meet (see here), the area of the handle is less than that of the other half. Any dividing line further in the handle would either give the handle less area or be significantly longer than the solutions available by restraining it to the square.

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    why haven't you taken in cosideration the area of the rectangle at the bottom? The segment could pass there too. – 2017-01-27
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    That's the one flaw: I "eyeballed" it and figured if it passed through there the areas wouldn't be equal. Working now to prove it rather than assume it. – 2017-01-27
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    This an interesting and highly intelligent approach to a grade 10 maths problem! However, looking back into the answers (as they come with the problem) it seems as though it wants a length in metres and a direction (either horizontal or vertical). I think it wants how far from the top left corner down, so I'm guessing it want y only? I've tried doing something similar with equations and will continue to look into it. Further ideas are much appreciated! – 2017-01-27
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    @mathsairhead Ah, I've re-read the problem and we seem to have missed a crucial part of the problem: "either north-south or east-west." They want only the vertical or horizontal cut. With that, should just be guess and check, since it's not gonna be on that slope and the rest are just normal fences from one edge to the other. Probably gonna be 3000 if i had to guess. – 2017-01-27
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    I'd 100% guess 3000 as well. I don't know if this is against any rules on here but the actual answer is 2020.83m and horizontal. It does seem as though there's a formula to this. I'll keep looking to see what I can do. – 2017-01-27
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    @mathsairhead 2020 and horizontal... no way that cuts it into equal area parts. That's like right at the top! Keep me updated on what you figure out, now i'm curious what they're doing. – 2017-01-27
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    Also, let us move to [this chatroom](http://chat.stackexchange.com/rooms/52568/splitting-a-shape-up-into-equal-halves) to avoid spamming future users. We'll post back when with conclusions when we make them, lol. – 2017-01-27
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Okay, looks like I have found an answer almost immediately!

I'm going to be working in both ha (hectares) and metres, but I'll specify when I switch.

So! The area of this shape is 987.5 ha which is 9875000 metres. Exactly half of this is 493.75 ha (4937500 metres).

What we need to find is how far down or across we need to make a line that splits the area in half exactly. This needs to be done with both the length and width in order to see which amount is smaller, and thus more affordable.

So, if solving for a line that goes east to west, we already know that the width of the shape is 3000 metres, which can be x. We don't know the length of the line, however, which can be y. But we DO know that there is a triangle that has been removed from the overall full shape (originally a rectangle) with the area of 112.5 hectares (1125000 metres). With this, we create this equation (in metres):

3000y – 1125000 = 4937500

If we solve this equation:

3000y – 1125000 = 4937500

3000y = 6062500

y = 2020.83333 metres

... Which is what the answers said!

But then I became curious and decided to see what a line would be like from north to south. Instead of focusing on finding y, I focused on finding x. Exactly half of the length of the overall shape is 2500 metres. So, I changed the equation into this:

2500x – 1125000 = 4937500

And then solve:

2500x – 1125000 = 4937500

2500x = 6062500

x = 2425 metres

... Which is larger than the east to west line and therefore more money per linear metre, making the answer rightfully 2020.83m and horizontal as the cheaper option!

I got this answer by working backwards, by already using 2020.83 as our y.

2020.83 * 3000 = 6062490

6062490 - 1125000 = 4937490 (493.749... Which is 493.75)

Hopefully this is easy to understand! Thanks for the help @Vedvart1 ! Here is your explanation!

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    Interesting, my answer is actually wrong! The one case I forgot to include was the correct one, where the line runs from the left side to the right side. – 2017-01-27