The "Box" Problem & The "Goat" Problem

I'll draw some pictures to explain what I'm doing.

This square stands for my initial piece of cardboard, out of which I'll be cutting corner squares and folding up the sides.

The square's dimensions are x inches by x inches, but I don't know the value of x yet:

Now I have cut the corners out, leaving two-inch flaps on all four sides. I will be folding the sides up along those red lines interior to the original rectangle:

Now I have folded up the flaps to make a two-inch deep box. (Let's assume that I will be taping the corners to hold them together.)

I will be using x to stand for both the length and the width of the original square of cardboard in the computations that follow. The value of x is what I'm looking for.

Looking at the third picture above, I can see that the width of the bottom of my box will be x − 2 − 2; that is, the final width of the box will be the sheet's original x inches, minus two inches on either side because of the portions that I'll be losing to the flaps I'll be folding up. Then the width of the bottom of my box is going to be x − 4. By the same reasoning, the length will also be x − 4.

Since the depth of my box is 2 (that's the whole point of the two-inch-by-two-inch squares that I'm cutting out of the corners), I can write the formula for the volume of this box as:

(I've spelled things out above, complete with color, to make my meaning clear.)

I can ignore the extraneous negative result. Then x = 20 inches, and the answer is:

What formula could they be expecting me to use for this? How on earth would I answer this?

When in doubt, it can be helpful to draw a picture, so let's see what I can find:

The goat is tied at the upper right-hand corner with the dot. Now what? Well, on the nearest two sides, the goat can go this far:

This is three-quarters of a circle with radius 8 (being the full length of the rope). But the goat can walk around the far corners as far as the rope will allow. What does this add to the three-quarter circle?

Along the top side, five meters of rope will be stretched along the side, leaving another three meters in play for wrapping down along the left-hand side of the rectangle I first drew; along the right-hand side, four meters will have been used, leaving another four meters for wrapping along the bottom of the rectangle:

These new areas are both quarter-circles, one with radius 3 and the other with radius 4. And there will be no way for the goat to reach the opposite corner or for any overlap in grazing, because he's out of rope. So these three partial circles mark off the total grazing area.

I know the formula for the area of a circle. To find, say, the area of 3/4 of a circle, I'd just multiply the total area by 3/4. So the total area is:

I'm supposed to give my answer in terms of the nearest whole unit, so I'll plug the above into my calculator, and round. It turns out that the goat is able to graze: