1. **Problem statement:** We need to find the volume of a wedge cut from a circular cylinder of radius 4 by two planes. One plane is perpendicular to the cylinder's axis, and the other intersects the first at a 30° angle along a diameter. 2. **Understanding the geometry:** The cylinder has radius $r=4$. The first plane cuts the cylinder perpendicular to its axis, creating a circular cross-section of area $\pi r^2$. 3. **Volume of the full cylinder:** If the cylinder's height is $h$, its volume is $V=\pi r^2 h$. 4. **Wedge volume:** The second plane intersects the first at a 30° angle, slicing the cylinder along a diameter. This creates a wedge-shaped volume which is a fraction of the full cylinder. 5. **Formula for wedge volume:** The wedge volume is proportional to the angle between the planes. Since the angle is $30^\circ$, the wedge volume is $\frac{30}{360} = \frac{1}{12}$ of the full cylinder volume. 6. **Expressing wedge volume:** $$ V_{wedge} = \frac{1}{12} \times \pi r^2 h $$ 7. **Final answer:** The volume of the wedge is $$ V_{wedge} = \frac{1}{12} \pi \times 4^2 \times h = \frac{1}{12} \pi \times 16 \times h = \frac{4\pi h}{3} $$ This formula gives the wedge volume in terms of the cylinder height $h$.