Question

the width of a plastic storage box is 2 ft longer than the height. the length is 6 ft longer than the height. the volume is 64 ft³. answer the following questions.
what is the formula for the volume of a rectangular prism?
what variable expressions represent the length, height, and width?
what equation represents the volume of the plastic storage box?
what are the dimensions of the box?

a. ( v=(h + 6)(h + 2)(h + 6) ), where ( h ) is the height of the box
b. ( v=(h^2 + 6)(h^2 + 2)(h^2) ), where ( h ) is the height of the box
c. ( v = h + 8 ), where ( h ) is the height of the box

what are the dimensions of the box?
the plastic storage box measures (square) ft in width, (square) ft in length, and (square) ft in height

Explanation:

Step1: Recall rectangular prism volume formula

The volume \( V \) of a rectangular prism is given by the product of its length \( l \), width \( w \), and height \( h \):
$$V = l \times w \times h$$

Step2: Define variables for dimensions

Let \( h \) = height of the box (in ft).

• Width is 2 ft longer than height: \( w = h + 2 \)
• Length is 6 ft longer than height: \( l = h + 6 \)

Step3: Set up volume equation

Substitute the variable expressions into the volume formula, using the given volume \( V = 64 \, \text{ft}^3 \):
$$(h + 6)(h + 2)h = 64$$
Expand and rearrange to standard polynomial form:
$$h(h^2 + 8h + 12) = 64$$
$$h^3 + 8h^2 + 12h - 64 = 0$$

Step4: Solve for height \( h \)

Test positive integer values (since dimensions are positive):

• For \( h = 2 \):

$$2^3 + 8(2)^2 + 12(2) - 64 = 8 + 32 + 24 - 64 = 0$$
So \( h = 2 \, \text{ft} \) is a valid solution.

Step5: Calculate width and length

• Width: \( w = h + 2 = 2 + 2 = 4 \, \text{ft} \)
• Length: \( l = h + 6 = 2 + 6 = 8 \, \text{ft} \)

Answer:

1. Formula for volume of a rectangular prism: \( V = lwh \)
2. Variable expressions:

• Height: \( h \)
• Width: \( h + 2 \)
• Length: \( h + 6 \)

1. Volume equation for the box: \( V = (h + 6)(h + 2)h \) (where \( h \) is the height of the box)
2. Dimensions of the box: 4 ft in width, 8 ft in length, and 2 ft in height