1. **Stating the problem:** We need to find the total area and the moment of inertia about the center for a shape described by dimensions $14 \times 2$ on the top side and $6 \times 12 \times 1.5$ on the right side, resembling a square-like frame with notches. 2. **Understanding the shape:** The shape is a frame with symmetrical indentations (notches) on all four sides. We interpret the dimensions as follows: - The outer square has side length $14 \times 2 = 28$ units. - The notches on the sides have dimensions $6 \times 12 \times 1.5$; likely this means each notch is $6$ units wide, $12$ units deep, and $1.5$ units thick (thickness may relate to depth or extrusion). 3. **Calculating the total area:** - Calculate the area of the outer square: $$A_{outer} = 28 \times 28 = 784$$ - Calculate the area of one notch (assuming rectangular notch): $$A_{notch} = 6 \times 12 = 72$$ - There are 4 notches (one on each side), so total notch area: $$A_{notches} = 4 \times 72 = 288$$ - Total area of the frame (outer square minus notches): $$A_{total} = A_{outer} - A_{notches} = 784 - 288 = 496$$ 4. **Calculating the moment of inertia about the center:** - Moment of inertia for a square about its center: $$I_{outer} = \frac{1}{12} \times b \times h^3 = \frac{1}{12} \times 28 \times 28^3 = \frac{1}{12} \times 28 \times 21952 = 51253.33$$ - Moment of inertia for one notch (rectangle) about its own centroid: $$I_{notch,centroid} = \frac{1}{12} \times 6 \times 12^3 = \frac{1}{12} \times 6 \times 1728 = 864$$ - Distance from notch centroid to square center (assuming notch centered on side): half side minus half notch depth: $$d = \frac{28}{2} - \frac{12}{2} = 14 - 6 = 8$$ - Using parallel axis theorem for one notch: $$I_{notch} = I_{notch,centroid} + A_{notch} \times d^2 = 864 + 72 \times 8^2 = 864 + 72 \times 64 = 864 + 4608 = 5472$$ - Total moment of inertia for 4 notches: $$I_{notches} = 4 \times 5472 = 21888$$ - Moment of inertia of the frame: $$I_{total} = I_{outer} - I_{notches} = 51253.33 - 21888 = 29365.33$$ 5. **Final answers:** - Total area: $496$ units squared - Moment of inertia about the center: approximately $29365.33$ units to the fourth power These calculations assume the thickness $1.5$ does not affect the 2D area and inertia calculations or is uniform in the third dimension.