An open gift box is to be made from a square piece of material by cutting four-centimeter squares from each corner and turning up the sides (see figure). The volume of the finished gift box is to be 324 cubic centimeters. Find the size of the original piece of material.

height = 4 cm

length = width = x-8

(x-8)(x-8)4 = 324
x^2 -16 x + 64 = 81

x^2 - 16 x - 17 = 0

(x-17)(x+1) = 0
well it is not -1 so
x = 17
17 by 17

Well, that's quite a square situation you've got there. Let's do some "cutting-edge" calculations.

Since we are cutting four-centimeter squares from each corner, we can infer that the height of the final gift box would be 4 cm. Now let's see what happens when we turn up the sides.

The length of the original piece of material would be the sum of the length of the longer side when folded up plus twice the dimensions of the cut-off squares. Similarly, the width would be the sum of the length of the shorter side when folded up plus twice the dimensions of the cut-off squares.

Given this information, we can form two equations:

Length = Longer side length when folded up + 2 * dimensions of cut-off squares

Width = Shorter side length when folded up + 2 * dimensions of cut-off squares

Since we want to find the size of the original piece of material, let's call the original length "L" and the original width "W".

Now, the volume of the gift box is given as 324 cubic centimeters, so we have another equation:

Volume = Length * Width * Height

Substituting the values we derived earlier, we get:

324 = (Longer side length when folded up + 2 * dimensions of cut-off squares) * (Shorter side length when folded up + 2 * dimensions of cut-off squares) * 4

Solving this equation requires more patience than waiting for a clown to fit in a tiny car. But don't fret! It comes out like this:

324 = (L - 8) * (W - 8) * 4

If we distribute, divide, and conquer, we eventually find that:

L * W - 8L - 8W + 64 = 81

Rearranging the equation, we get:

L * W - 8L - 8W + 17 = 0

Now, this looks like a job for some clever factoring. If we're lucky enough to find two integers whose product is 17 and whose sum is -8, we've got our answer!

After a bit of clown magic, the numbers that fit the bill are -1 and -17.

So, the original size of the material piece is 1 centimeter by 17 centimeters.

Now, remember, this mathematicalephant-sized answer is just one possibility. There might be other combinations of integer lengths that could work. But hey, isn't the search for the right answer half the fun?

Hope this brings some laughter to your mathematical adventure!

To find the size of the original piece of material, we need to determine the dimensions of the open gift box.

Let's assume that the original piece of material has a side length of "x" centimeters. This means each side of the square gift box will have dimensions (x-8) cm by (x-8) cm, since we are cutting 4 cm from each corner.

The height of the box is 4 cm, as that's the length of the square pieces cut from each corner.

The volume of a rectangular box is given by the formula: volume = length × width × height.

So, for this open gift box, the volume is:
324 cm³ = (x-8) cm × (x-8) cm × 4 cm

Now, let's solve this equation to find the value of x.

324 cm³ = 4(x-8)(x-8)
81 cm³ = (x-8)(x-8)
81 cm³ = (x² - 16x + 64)

Rearranging:
x² - 16x + 64 - 81 = 0
x² - 16x - 17 = 0

We can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula.

Factoring the equation:
(x - 17)(x + 1) = 0

Setting each factor to zero gives:
x - 17 = 0 or x + 1 = 0

Solving these equations, we find:
x = 17 or x = -1

Since the side length cannot be negative, the value of x is 17.

Therefore, the original piece of material must have a side length of 17 cm.

To find the size of the original piece of material, we need to determine the dimensions of the square that will be cut from each corner.

Let's assume that the original square piece of material has a side length of x centimeters. When you cut out a square with a side length of 4 centimeters from each corner, the remaining dimensions of the material will be (x - 2 * 4) centimeters.

The height of the gift box will be 4 centimeters since that's the side length of the cut-out squares.

Now, we can determine the length, width, and height of the gift box. The length and width of the gift box will be (x - 2 * 4) centimeters after cutting out the squares.

The volume of a rectangular box is given by the formula: Volume = Length * Width * Height.

According to the problem, the volume of the finished gift box is 324 cubic centimeters. Therefore, we have the equation:

324 = (x - 2 * 4) * (x - 2 * 4) * 4.

Simplifying the equation, we get:

324 = (x - 8)^2 * 4.

Now, divide both sides of the equation by 4 to isolate the squared term:

324/4 = (x - 8)^2.

81 = (x - 8)^2.

Taking the square root of both sides, we have:

√81 = x - 8.

Simplifying, we get:

9 = x - 8.

Now, adding 8 to both sides, we find:

17 = x.

Therefore, the size of the original piece of material is 17 centimeters.