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.

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.