Let's break down the problem using the given values: - \( a = 12 \) in. - \( \theta = 45^\circ \) The moments of inertia with respect to the \(x\) and \(y\) axes are typically given (for a composite or built-up section) by summing the individual moments of inertia of each rectangle about the centroid, plus the parallel axis theorem as needed. ### 1. Moments of Inertia about \( x \) and \( y \) axes: For a rectangle: - About base: \( I_{\text{base}} = \frac{1}{12} b h^3 \) - About side: \( I_{\text{side}} = \frac{1}{12} h b^3 \) Let's assume you have already calculated \( I_x \) and \( I_y \). ### 2. Moments of Inertia about \( u \) and \( v \) axes: The axes \( u \) and \( v \) are rotated by \( \theta = 45^\circ \): The transformed moments of inertia are: \[ I_u = I_x \cos^2\theta + I_y \sin^2\theta - 2I_{xy} \sin\theta\cos\theta \] \[ I_v = I_x \sin^2\theta + I_y \cos^2\theta + 2I_{xy} \sin\theta\cos\theta \] \[ I_{uv} = (I_y - I_x) \sin\theta\cos\theta + I_{xy} (\cos^2\theta - \sin^2\theta) \] ### 3. Plug in the values: For \( \theta = 45^\circ \): - \( \sin\theta = \cos\theta = \frac{\sqrt{2}}{2} \) - \( 2\sin\theta\cos\theta = \sin(2\theta) = 1 \) \[ I_u = I_x \left(\frac{1}{2}\right) + I_y \left(\frac{1}{2}\right) - I_{xy} \] \[ I_v = I_x \left(\frac{1}{2}\right) + I_y \left(\frac{1}{2}\right) + I_{xy} \] \[ I_{uv} = (I_y - I_x) \left(\frac{1}{2}\right) \] ### 4. You need \( I_x \), \( I_y \), and \( I_{xy} \): Assume you have already calculated (for the composite section): - \( I_x \) - \( I_y \) - \( I_{xy} \) ### 5. Substitute and Calculate: Let's say, for example: - \( I_x = A \) - \( I_y = B \) - \( I_{xy} = C \) Then: \[ I_u = \frac{A + B}{2} - C \] \[ I_v = \frac{A + B}{2} + C \] \[ I_{uv} = \frac{B - A}{2} \] --- #### **To get the actual numbers, you need to calculate the moments of inertia for the given T-section (shown in the figure) using the parallel axis theorem.** If you provide the calculations for \( I_x \), \( I_y \), and \( I_{xy} \) or their values, I can plug them in for the final answers. Would you like to continue with the actual calculations for the T-section cross-section shown?