Question

A company wants to design a series of boxes in the shape of cubes. The volume of each cube cannot exceed 1,728 cubic inches. The volume of a box can be represented by the function y=x3, where x represents the possible side lengths of the box, in inches. What is the reasonable domain and range of the polynomial function?(1 point)
Responses

Domain: (0,∞);

Range: (0,1,728]

Domain: open paren 0 comma infinity close paren ; Range: left parenthesis 0 comma 1 comma 728 right bracket

Domain: (0,12];

Range: (0,∞)

Domain: left parenthesis 0 comma 12 right bracket ; Range: open paren 0 comma infinity close paren

Domain: (−∞,∞);

Range: (−∞,∞)

Domain: open paren negative infinity comma infinity close paren ; Range: open paren negative infinity comma infinity close paren

Domain: (0,12];

Range: (0,1,728]

Answer 1

To find the reasonable domain and range of the function \( y = x^3 \) given that the volume cannot exceed 1,728 cubic inches, we need to determine the constraints on the side length \( x \) of the cube.

1. **Determine the maximum side length:**
Since the volume \( V = x^3 \) cannot exceed 1,728 cubic inches, we have the inequality:
\[
x^3 \leq 1728
\]
To find the maximum value of \( x \), we take the cube root of both sides:
\[
x \leq \sqrt[3]{1728} = 12
\]

2. **Determine the domain:**
The side length \( x \) must also be greater than \( 0 \) because lengths cannot be negative or zero. Therefore, the reasonable domain of the function is:
\[
(0, 12]
\]

3. **Determine the range:**
The smallest value of \( y \) occurs when \( x \) is at its minimum value (since \( x > 0 \)), which will approach \( 0 \) as \( x \) approaches \( 0 \). The largest value of \( y \) occurs when \( x = 12 \):
\[
y = 12^3 = 1728
\]
Thus, the range of \( y \) is:
\[
(0, 1728]
\]

Combining these findings, we get:
- **Domain:** \( (0, 12] \)
- **Range:** \( (0, 1728] \)

Given this analysis, the correct response is:
**Domain: (0, 12]; Range: (0, 1,728]**