Question

the width of a plastic storage box is 3 ft longer than the height. the length is 7 ft longer than the height. the volume is 308 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? options: c. v = (h + 7)(h + 3)(h + 10), where h is the height of the box; d. v = h + 10, where h is the height of the box. what are the dimensions of the box? the plastic storage box measures □ ft in width, □ ft in length, and □ ft in height.

Explanation:

Step1: State volume formula

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

Step2: Define variable expressions

Let height = $h$. Width is 3 ft longer than height: $w = h+3$. Length is 7 ft longer than height: $l = h+7$.

Step3: Set up volume equation

Substitute expressions into volume formula:
$V = h(h+3)(h+7)$

Step4: Solve for height $h$

Substitute $V=308$:
$$h(h+3)(h+7) = 308$$
Expand left side:
$$h(h^2+10h+21) = 308$$
$$h^3+10h^2+21h-308=0$$
Test integer values: $h=4$ works:
$4^3+10(4)^2+21(4)-308=64+160+84-308=0$

Step5: Calculate dimensions

Height $h=4$ ft. Width: $4+3=7$ ft. Length: $4+7=11$ ft.
Verify: $4×7×11=308$, which matches the given volume.

Answer:

1. $V = lwh$
2. Height: $h$, Width: $h+3$, Length: $h+7$
3. $V = h(h+3)(h+7)$
4. 7 ft (width), 11 ft (length), 4 ft (height)