1. **Problem statement:** We are given a cuboid with base rectangle PQRS and top face W T U V. PR is the diagonal of the base, and M is the midpoint of PR. PU is a vertical edge, and the angle between PU and the base plane PQRS is 30°. We need to find the size of angle $\angle UMR$ correct to 3 significant figures. 2. **Understanding the problem:** - PR = 42 cm. - M is midpoint of PR, so $PM = MR = \frac{42}{2} = 21$ cm. - PU is vertical, so PU is perpendicular to the base plane PQRS. - The angle between PU and the plane PQRS is 30°, meaning PU makes a 30° angle with the base plane. 3. **Key formulas and concepts:** - Since PU is vertical, the length of PU can be related to the angle 30° using trigonometry. - The angle between PU and the plane is 30°, so the vertical height $PU = PR \times \tan 30^\circ$. - Coordinates can be assigned to points to find vectors and then the angle $\angle UMR$. 4. **Assign coordinates:** - Let P = (0,0,0). - Since PR is diagonal of base rectangle, let R = (42,0,0) along x-axis. - M is midpoint of PR: $M = (21,0,0)$. - PU is vertical edge from P, so U = (0,0,h) where $h = PU$. 5. **Find height $h$:** - The angle between PU and base plane is 30°, so $$\tan 30^\circ = \frac{PU}{PR} = \frac{h}{42} \implies h = 42 \times \tan 30^\circ$$ - Calculate $\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.57735$ - So, $$h = 42 \times 0.57735 = 24.2487 \text{ cm}$$ 6. **Find vectors UM and MR:** - U = (0,0,24.2487) - M = (21,0,0) - R = (42,0,0) - Vector $\overrightarrow{UM} = M - U = (21 - 0, 0 - 0, 0 - 24.2487) = (21, 0, -24.2487)$ - Vector $\overrightarrow{MR} = R - M = (42 - 21, 0 - 0, 0 - 0) = (21, 0, 0)$ 7. **Calculate angle $\angle UMR$ between vectors $\overrightarrow{UM}$ and $\overrightarrow{MR}$:** - Use dot product formula: $$\overrightarrow{UM} \cdot \overrightarrow{MR} = |UM||MR| \cos \theta$$ - Calculate dot product: $$21 \times 21 + 0 + 0 = 441$$ - Calculate magnitudes: $$|UM| = \sqrt{21^2 + 0 + (-24.2487)^2} = \sqrt{441 + 588.04} = \sqrt{1029.04} = 32.08$$ $$|MR| = \sqrt{21^2 + 0 + 0} = 21$$ - Calculate $\cos \theta$: $$\cos \theta = \frac{441}{32.08 \times 21} = \frac{441}{673.68} = 0.6547$$ - Find $\theta$: $$\theta = \cos^{-1}(0.6547) = 49.1^\circ$$ 8. **Final answer:** The size of angle $\angle UMR$ is approximately $\boxed{49.1^\circ}$ to 3 significant figures.