Question Construct an open-top box from a $12 \mathrm{~cm} x 16 \mathrm{~cm}$ sheet. The volume $v(x)$ is a function of the side length $x$ . Which expression represents $v(x)$ ?

Studdy Solution

STEP 1

Assumptions
1. The original sheet of paper is $12 \text{ cm} \times 16 \text{ cm}$ in size.
2. A square with side length $x$ is cut out from each corner of the sheet.
3. The box is formed by folding up the sides along the cuts.
4. The volume $v(x)$ of the box is a function of the side length $x$ of the cutout.
5. The box has an open top, so it has no lid.

STEP 2

To find the expression for the volume $v(x)$ , we need to determine the new dimensions of the box after the squares are cut out and the sides are folded up.

STEP 3

After cutting out the squares, the new length of the box will be the original length minus two times the side length of the square cutout.
$New\, length = Original\, length - 2x$

STEP 4

Plug in the original length of the paper to find the new length of the box.
$New\, length = 16 \text{ cm} - 2x$

STEP 5

Similarly, the new width of the box will be the original width minus two times the side length of the square cutout.
$New\, width = Original\, width - 2x$

STEP 6

Plug in the original width of the paper to find the new width of the box.
$New\, width = 12 \text{ cm} - 2x$

STEP 7

The height of the box will be equal to the side length $x$ of the square cutout, as this is the portion that is folded up to form the sides.
$Height = x$

STEP 8

The volume of a box is given by the product of its length, width, and height.
$v(x) = Length \times Width \times Height$

STEP 9

Substitute the expressions for the new length, new width, and height into the volume formula.
$v(x) = (16 \text{ cm} - 2x) \times (12 \text{ cm} - 2x) \times x$

STEP 10

Simplify the expression by distributing the $x$ and combining like terms if necessary.
$v(x) = x(16 - 2x)(12 - 2x)$

STEP 11

Now we compare the simplified expression for $v(x)$ with the given options to find the correct one.
The correct expression for the volume $v(x)$ is:
$v(x) = (12 - 2x)(16 - 2x)x$
This matches the second option given in the problem statement.
Therefore, the expression that represents $v(x)$ is $V(x) = (12 - 2x)(16 - 2x)x$ .

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Question Construct an open-top box from a $12 \mathrm{~cm} x 16 \mathrm{~cm}$ sheet. The volume $v(x)$ is a function of the side length $x$ . Which expression represents $v(x)$ ?

Studdy Solution

STEP 1

STEP 2

To find the expression for the volume $v(x)$ , we need to determine the new dimensions of the box after the squares are cut out and the sides are folded up.

STEP 3

After cutting out the squares, the new length of the box will be the original length minus two times the side length of the square cutout.
$New\, length = Original\, length - 2x$

STEP 4

Plug in the original length of the paper to find the new length of the box.
$New\, length = 16 \text{ cm} - 2x$

STEP 5

Similarly, the new width of the box will be the original width minus two times the side length of the square cutout.
$New\, width = Original\, width - 2x$

STEP 6

Plug in the original width of the paper to find the new width of the box.
$New\, width = 12 \text{ cm} - 2x$

STEP 7

The height of the box will be equal to the side length $x$ of the square cutout, as this is the portion that is folded up to form the sides.
$Height = x$

STEP 8

The volume of a box is given by the product of its length, width, and height.
$v(x) = Length \times Width \times Height$

STEP 9

Substitute the expressions for the new length, new width, and height into the volume formula.
$v(x) = (16 \text{ cm} - 2x) \times (12 \text{ cm} - 2x) \times x$

STEP 10

Simplify the expression by distributing the $x$ and combining like terms if necessary.
$v(x) = x(16 - 2x)(12 - 2x)$

STEP 11

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Question Construct an open-top box from a 12 cmx16 cm12 \mathrm{~cm} x 16 \mathrm{~cm}12 cmx16 cm sheet. The volume v(x)v(x)v(x) is a function of the side length xxx. Which expression represents v(x)v(x)v(x)?

Studdy Solution

STEP 1

STEP 2

To find the expression for the volume v(x)v(x)v(x), we need to determine the new dimensions of the box after the squares are cut out and the sides are folded up.

STEP 3

After cutting out the squares, the new length of the box will be the original length minus two times the side length of the square cutout.New length=Original length−2xNew\, length = Original\, length - 2xNewlength=Originallength−2x

STEP 4

Plug in the original length of the paper to find the new length of the box.New length=16 cm−2xNew\, length = 16 \text{ cm} - 2xNewlength=16 cm−2x

STEP 5

Similarly, the new width of the box will be the original width minus two times the side length of the square cutout.New width=Original width−2xNew\, width = Original\, width - 2xNewwidth=Originalwidth−2x

STEP 6

Plug in the original width of the paper to find the new width of the box.New width=12 cm−2xNew\, width = 12 \text{ cm} - 2xNewwidth=12 cm−2x

STEP 7

The height of the box will be equal to the side length xxx of the square cutout, as this is the portion that is folded up to form the sides.Height=xHeight = xHeight=x

STEP 8

The volume of a box is given by the product of its length, width, and height.v(x)=Length×Width×Heightv(x) = Length \times Width \times Heightv(x)=Length×Width×Height

STEP 9

Substitute the expressions for the new length, new width, and height into the volume formula.v(x)=(16 cm−2x)×(12 cm−2x)×xv(x) = (16 \text{ cm} - 2x) \times (12 \text{ cm} - 2x) \times xv(x)=(16 cm−2x)×(12 cm−2x)×x

STEP 10

Simplify the expression by distributing the xxx and combining like terms if necessary.v(x)=x(16−2x)(12−2x)v(x) = x(16 - 2x)(12 - 2x)v(x)=x(16−2x)(12−2x)

STEP 11

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Question Construct an open-top box from a $12 \mathrm{~cm} x 16 \mathrm{~cm}$ sheet. The volume $v(x)$ is a function of the side length $x$ . Which expression represents $v(x)$ ?

To find the expression for the volume $v(x)$ , we need to determine the new dimensions of the box after the squares are cut out and the sides are folded up.

After cutting out the squares, the new length of the box will be the original length minus two times the side length of the square cutout.
$New\, length = Original\, length - 2x$

Plug in the original length of the paper to find the new length of the box.
$New\, length = 16 \text{ cm} - 2x$

Similarly, the new width of the box will be the original width minus two times the side length of the square cutout.
$New\, width = Original\, width - 2x$

Plug in the original width of the paper to find the new width of the box.
$New\, width = 12 \text{ cm} - 2x$

The height of the box will be equal to the side length $x$ of the square cutout, as this is the portion that is folded up to form the sides.
$Height = x$

The volume of a box is given by the product of its length, width, and height.
$v(x) = Length \times Width \times Height$

Substitute the expressions for the new length, new width, and height into the volume formula.
$v(x) = (16 \text{ cm} - 2x) \times (12 \text{ cm} - 2x) \times x$

Simplify the expression by distributing the $x$ and combining like terms if necessary.
$v(x) = x(16 - 2x)(12 - 2x)$