1. **Problem statement:** We need to find the angle $\theta$ between the base and the slant diagonal along the width and height of a cuboid with dimensions 17 mm (length), 10 mm (width), and 3 mm (height). 2. **Understanding the problem:** The angle $\theta$ is formed between the base (width side) and the slant diagonal that runs along the width and height. 3. **Step 1: Identify the triangle involved.** The triangle is formed by the width (10 mm), height (3 mm), and the slant diagonal (hypotenuse) between these two. 4. **Step 2: Calculate the slant diagonal length $d$ using Pythagoras' theorem:** $$d = \sqrt{10^2 + 3^2} = \sqrt{100 + 9} = \sqrt{109}$$ 5. **Step 3: Calculate angle $\theta$ using the cosine rule or trigonometric ratio.** Since $\theta$ is between the base (width) and the slant diagonal, use cosine: $$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{10}{\sqrt{109}}$$ 6. **Step 4: Calculate $\theta$:** $$\theta = \cos^{-1}\left(\frac{10}{\sqrt{109}}\right)$$ 7. **Step 5: Evaluate numerically:** $$\sqrt{109} \approx 10.4403$$ $$\frac{10}{10.4403} \approx 0.9571$$ $$\theta \approx \cos^{-1}(0.9571) \approx 16.5^\circ$$ **Final answer:** $$\boxed{16.5^\circ}$$