Question

jenny uses small boxes to ship the earrings she sells on her jewelry website. the box is in the shape of a rectangular prism and has a volume of 8 cubic inches. the length of the box is 2 inches. the height of the box is 3 inches more than the width. complete the equation that models the volume of the box in terms of the width, \\(x\\).

volume of a prism \\((v) = (\text{length})(\text{width})(\text{height})\\)

choose an answer \\(x^2 +\\) choose an answer \\(x =\\) choose an answer

it choose an answer possible for the width of this box to be 1.5 inches.

Explanation:

Define variables and dimensions

Using the Algebraic Translation and Formulating Equations knowledge points:

• Let the width of the box be \(x\) (in inches).
• The length of the box is given as \(2\) inches.
• The height of the box is \(3\) inches more than the width, which is \(x + 3\).

Formulate the volume equation

Using the Formulating Equations knowledge point:

• The formula for the volume of a rectangular prism is:

$$V = \text{length} \times \text{width} \times \text{height}$$

• Substitute the given values and expressions:

$$8 = 2 \cdot x \cdot (x + 3)$$

• Expand the right side of the equation:

$$8 = 2x(x + 3)$$

$$8 = 2x^2 + 6x$$

• Rearrange to match the template \(\text{[ ]}x^2 + \text{[ ]}x = \text{[ ]}\):

$$2x^2 + 6x = 8$$

Analyze the feasibility of a 1.5-inch width

Using the Formulating Equations knowledge point:

• Substitute \(x = 1.5\) into the left side of our volume equation:

$$2(1.5)^2 + 6(1.5) = 2(2.25) + 9 = 4.5 + 9 = 13.5$$

• Since \(13.5

eq 8\), a width of \(1.5\) inches does not satisfy the volume requirement.

• Therefore, it is not possible for the width to be \(1.5\) inches.

Answer:

Question 1

<blank>2</blank> \(x^2 +\) <blank>6</blank> \(x =\) <blank>8</blank>

Question 2

It <blank>is not</blank> possible for the width of this box to be 1.5 inches.