Question

A box is formed by cutting squares from the four corners of a 9"-wide by 12"-long sheet of paper and folding up the sides.
Let x represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches).
Write a formula that expresses V in terms of x.

Answer 1

Answer:

$V(x) = 4x^3-42x^2 +108x$

Step-by-step explanation:

We are given the following in the question:

Width of sheet = 9 inches

Length of sheet = 12 inches

A square of length x inches is cut from each corner of the sheet to form a box.

Thus, the dimension of the box formed is:

Height = x inches

Width =

$9 - x-x = (9-2x)\text{ inches}$

Length =

$12 - x-x = (12-2x)\text{ inches}$

Volume of box formed =

$\text{Length}\times \text{Width}\times \text{Height}$

Putting values, we get,

$V(x) = (12-2x)(9-2x)x\\V(x) = (12-2x)(9x-2x^2)\\V(x) = 108x-24x^2-18x^2+4x^3\\V(x) = 4x^3-42x^2 +108x$

is the required expression of volume of box in terms of x.

Answer 2

The volume of a box is the amount of space in it.

The expression that represents volume is: $\mathbf{V = (12 -2x) (9 - 2x)x}$

The dimension of the cardboard is given as:

$\mathbf{Length = 12}$

$\mathbf{Width = 9}$

Assume the cut-out is x.

So, the dimension of the box is:

$\mathbf{Length =12-2x}$

$\mathbf{Width =9 - 2x}$

$\mathbf{Height = x}$

The volume of the box is:

$\mathbf{V = (12 -2x) (9 - 2x)x}$

Hence, the expression that represents volume is:

$\mathbf{V = (12 -2x) (9 - 2x)x}$

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