10. The volume of a cube is a function of its side length. Similarly, the surface area of a cube is a function of its side length. Which of these relationships is a linear function? Construct an argument to justify your response.

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Neither the volume nor the surface area of a cube is a linear function of its side length. Here's why:

### Understanding Linear Functions
A **linear function** generally has the form:
$$ f(x) = mx + b $$
where:
- $$ m $$ is the slope (constant rate of change),
- $$ b $$ is the y-intercept.

In a linear function, the graph is a straight line, and the relationship between the variables involves no exponents other than 1.

### Volume of a Cube
The volume $$ V $$ of a cube with side length $$ s $$ is given by:
$$ V(s) = s^3 $$
- **Nature of the Function:** This is a **cubic function** because the side length $$ s $$ is raised to the third power.
- **Implications:** The rate at which the volume changes with respect to the side length increases rapidly as $$ s $$ increases. The graph of $$ V(s) $$ versus $$ s $$ is a curve, not a straight line.

### Surface Area of a Cube
The surface area $$ A $$ of a cube with side length $$ s $$ is given by:
$$ A(s) = 6s^2 $$
- **Nature of the Function:** This is a **quadratic function** because the side length $$ s $$ is squared.
- **Implications:** The surface area increases proportionally to the square of the side length. Like the volume, the graph of $$ A(s) $$ versus $$ s $$ is a curve, not a straight line.

### Conclusion
Both functions involve exponents greater than one:
- **Volume:** $$ s^3 $$ (cubic)
- **Surface Area:** $$ s^2 $$ (quadratic)

Since neither function fits the form of a linear function ($$ f(x) = mx + b $$), **neither the volume nor the surface area of a cube is a linear function** of its side length.

**Summary:** Both the cube’s volume (proportional to s³) and surface area (proportional to s²) change nonlinearly with side length. Thus, neither relationship is a linear function.
Neither the volume nor the surface area of a cube is a linear function of its side length because both involve exponents greater than one.

The volume of a cube is a function of its
side length. Similarly, the surface area of
a cube is a function of its side length.
Which of these relationships is a linear
function? Construct an argument to justify
your response.

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