Final answer:
The correct moment of inertia of a triangle with respect to its base, not given in the options, is 1/18 × base × height². For the area calculation of a triangle with a base of 166 mm and height of 930 mm, the area is 0.0772 m² to four significant figures.
Step-by-step explanation:
The moment of inertia of a triangle with respect to its base is described by the formula 1/12 × base × height². This formula is derived from integrating the contributions of infinitesimal elements across the area of the triangle, taking into account their distance from the axis of rotation along the base of the triangle. Among the given options, none exactly represent the correct formula.
Using the parallel axis theorem, if we already know the moment of inertia with respect to an axis through the center of mass of the triangle and parallel to its base (which is 1/36 × base × height²), we can shift this axis to the base to find the moment of inertia about the base as 1/18 × base × height². It's important to note that none of the provided options matches this correct value.
To answer the related question regarding the area of a triangle, we can use the formula 1/2 × base × height. For a triangle with a base of 166 mm and a height of 930.0 mm, the area in square millimeters would be 1/2 × 166 mm × 930.0 mm, which simplifies to 77,190 mm². Since we are asked to express the area in square meters and also in proper significant figures, we need to convert the units (1 mm² = 1×10⁻¶ m²) and round accordingly, giving an area of 0.0772 m² to four significant figures.