Area properties – rectangular section about centroidal X–X axis Compute the second moment of area (moment of inertia) about the X–X axis for a rectangle that is 3 cm wide (b = 3 cm) and 4 cm deep (h = 4 cm), taking X–X through the centroid and parallel to the base.

Introduction / Context:The second moment of area (area moment of inertia) of a plane section is a geometric property used in bending, deflection, and stability calculations. For standard shapes, compact formulas exist. Here we determine I about the centroidal horizontal axis X–X for a rectangular section.

Given Data / Assumptions:

• Width b = 3 cm (horizontal).
• Depth/height h = 4 cm (vertical).
• X–X axis passes through the centroid and is parallel to the base (horizontal axis).
• Use standard formula I_x = b * h^3 / 12.

Concept / Approach:For a rectangle, the centroidal second moment about the axis parallel to its base is I_x = b * h^3 / 12. This comes from integrating y^2 dA over the area. Units are in length^4.

Step-by-Step Solution:

Verification / Alternative check:Dimensional check: input lengths in cm yield I in cm^4. Also note symmetry: increasing depth strongly increases I because of the h^3 factor, consistent with 16 cm^4 being larger than values using smaller h.

Why Other Options Are Wrong:

• 9 cm^4 or 12 cm^4: arise from using b^3 h / 12 or arithmetic slips.
• 20 cm^4: not supported by the formula; likely from averaging or misusing 1/10 instead of 1/12.

Common Pitfalls:Confusing axes: I_x uses h^3, whereas I_y about the vertical centroidal axis would be h * b^3 / 12. Always align the cubic dimension with the axis perpendicular to bending.

Final Answer:16 cm^4