A box with no top is to be constructed from a piece of cardboard whose length measures 12 inches more than its width. the box is formed by cutting squares that measures 4 inches on each sides from 4 corners and then folding up the sides. If the volume of the box will be 340 inches to the third, what are the dimensions of the piece of cardboard? Width? Length?

inches to the third are called cubic inches!

the original dimensions of the cardboard are w and w+12

The base of the box is thus
w-8 by w+4

Since the height of the box is now 4, the volume is

(w-8)(w+4)(4) = 340

solve that to get w=13 and the rest is downhill

(29-2x)(15-2x)

Let's start by finding the dimensions of the box.

Let's assume the width of the piece of cardboard is "x" inches.

According to the given information, the length of the cardboard is 12 inches more than its width, so the length is "x + 12" inches.

To find the dimensions of the box, we need to subtract the length of the square cutouts from the dimensions of the cardboard.

Each square cutout measures 4 inches on each side, so the length and width of the box will be reduced by 4 inches on each side.

The width of the box will be (x - 4) inches, and the length will be (x + 12 - 4) inches or (x + 8) inches.

The height of the box will be 4 inches since we are folding the sides up.

Now we can find the volume of the box using the formula:
Volume = Length * Width * Height.

Given that the volume is 340 inches³, we have the equation:
340 = (x + 8) * (x - 4) * 4.

Simplifying the equation, we get:
340 = 4(x + 8)(x - 4).

Divide both sides of the equation by 4:
85 = (x + 8)(x - 4).

Expand and rearrange the equation:
85 = (x² + 8x) - (4x + 32).
85 = x² + 8x - 4x - 32.
85 = x² + 4x - 32.

Move all terms to one side of the equation:
x² + 4x - 117 = 0.

Now we can solve this quadratic equation for x.

Factoring this equation or using the quadratic formula, we find that x ≈ 7.

Therefore, the width of the cardboard is approximately 7 inches.

The length of the cardboard is given as (x + 12), so it would be (7 + 12) = 19 inches.

So, the dimensions of the piece of cardboard are width = 7 inches and length = 19 inches.

To find the dimensions of the cardboard, we need to set up an equation based on the given information.

Let's assume the width of the cardboard is "w" inches. According to the given information, the length of the cardboard is 12 inches more than its width, so the length would be "w + 12" inches.

To create the box, we need to remove squares measuring 4 inches on each side from all four corners. This means the dimensions of the box will be reduced by twice the length of the square, which is 4 inches. Therefore, the dimensions of the box would be (w - 8) inches for the width and (w + 12 - 8) inches for the length.

The volume of a rectangular box can be calculated by multiplying the length, width, and height. In this case, the height of the box is 4 inches. So, we can set up the equation:

(w - 8) * (w + 12 - 8) * 4 = 340

Simplifying the equation, we get:

(w - 8) * (w + 4) = 85

Expanding the equation:

w^2 + 4w - 8w - 32 = 85

Combining like terms, we have:

w^2 - 4w - 117 = 0

Now, we can solve this quadratic equation to find the value of 'w'.

Using factoring, completing the square or the quadratic formula, we find:

(w - 13)(w + 9) = 0

This gives us two possible values for 'w': w = 13 or w = -9.

Since width cannot be negative, we ignore the negative value. Therefore, the width of the cardboard is 13 inches.

We can substitute this value back into the equation for the length:

Length = w + 12 = 13 + 12 = 25 inches

So, the dimensions of the piece of cardboard are: Width = 13 inches and Length = 25 inches.