Let me give a bit of background first.

At cats protection we use a product called Anistel (brand name) to disinfect damn near everything. From the litter trays to the food bowls to the floor itself. It has to be in a ratio of 1:100 (with water), undiluted it can cause lesions on the skin, both of humans and of cats!

We have an approximately cuboid box which we keep the bag of used cat litter in. This has two purposes, if any litter is spilled when emptying the litter boxes it goes in the box not on the floor. And it can be closed to keep the odours in the box.

(image credit: http://www.gallifreypermaculture.com.au/wp-content/gallery/misc_post_pics/plasticcrate.jpg)

As you can see it's slightly wider and longer at the top than at the bottom.

So when going to clean the box which had little bits of cat litter in it, I need to know the volume in litres in order to work out the dilution. This is fairly simple, as it has straight sides to treat it like a cuboid box but take the mean of the width of the top and the width of the base and the length of the top and of the bottom. Work these out you get:

46cm x 33 cm x 39cm

= 52026cm³

Divide by 1000 to convert to litres gets a nice handy 52.026

Bearing in my the ratio is 1:100 not 1:99 you need to divide by 101 to get the 1 part and obviously multiply that by 100 to get the 100 part.

Now this is where I've got stuck so far. In order to not fill it right to the brim, why not put exactly 50 litres of water in and add 0.5l of Anistel? The only thing I can vary is the height (39cm) as I can't change any of the dimensions of the box, and I can only fill it with it the right way up without the water just following straight out.

Of course this isn't linear so I can't just multiply the height by 50/52.026. Admittedly in practical terms this is close enough as the ration doesn't have to be exactly 1:100 but just fairly close to it. We don't measure to the millilitre. But since it's not linear I need to do something else.

Let's give all the numbers first, the volume happens to be (50+42)/2 x (37+29)/2 x 39. 39 is the perpendicular height in cm. But of course, if we don't go all the way to the top, the top length and width are not 50cm and 37cm anymore!

So I thought I'd start with a two dimensional model. It is to scale. At this point it's best to stop using the actual numbers and use algebra to solve the problem for all cases:

So, using this notation the area of the whole thing is (2a + 2b)/2 x h which is just h(a + b)

If we reduce h, a always stays the same but b is reduced by the same coefficient. Might as well use x for the decrease in h, which gives us

(h-x) x (2a + [2(b[h-x]/h)])/2

The bracketing is particularly ugly which is why I've alternated between parentheses and square brackets.

Basically because of how similar triangles work, the coefficient of the decrease of h which is (h-x)/h, we then have to multiply that by b to find out what the new b is. To state the obvious, h>x, x>0, a>0, b>0 (basically everything is more than zero and h is greater than x).

This is as far as you can go without using the actual numbers, because to turn it into an equation, something has to be equal to something else.

So, all in cm, a = 42, b = 4, h = 39

Area of the whole thing is 46 x 39 = 1794 cm²

This is why we need real numbers. I haven't specific x because it depends on what the area you want is. Note that if you use the longer formula (h-x) x (2a + [2(b[h-x]/h)/2] where x=0 you end up with the same answer. So if I want to find out what value of x and therefore of h-x I want for the area to be 1500cm

1500 = (39 - x) x (84 + [2(4[39 - x]/39)])/2

Must take a break. To be continued...

Alright, meant to take a break but actually just continued on pen and paper

Get rid of the final /2:

3000 = (39 - x) × (84 + [2(4[39 - x]/39)])

3000 = (39 - x) × (84 + [8(39 - x)/39)])

3000 = (39 - x) × (84 + [316 - 8x]/39)

This is as simplified as it gets before the painful multiplying the brackets stage:

3000 = 3276 - 84x + (12324 - 312x - 316x + 8x²)/39

Collect the exes in the brackets, then multiply everything by 36:

11700 = 127764 - 3276x + 12324 - 628x + 8x²

More collecting of like terms:

11700 = 140088 - 3904x + 8x²

Another break, with an actual break this time

Take 140088 from each side, then multiply by -1 (flip the signs as it's usually known)

128388 = 3904x - 8x²

Divide everything by 8

16048.5 = 488x - x²

And I've definitely gone wrong. I don't know how to solve it, but I know x>16048.5/488, which means x>32.89 which is clearly bollocks since we started with x<h we know h-x can't be as little as 6.11 which is less than 6.11/39 of the original area.
As a juggler, I won't delete this but keep it because all the time you spending getting it wrong makes it sweeter when you get it right.
Martin