1. **Problem statement:** We have a greenhouse framework with a cross-section shaped as a quarter-circle of radius 4 metres. The length of the greenhouse (depth) is 8 metres. We need to calculate the minimum total length of wood needed to build the framework. 2. **Understanding the framework:** - The cross-section is a quarter-circle with radius $r=4$ m. - The length of the greenhouse (depth) is $8$ m. - The framework consists of the quarter-circle arc plus the length of the greenhouse along the depth. 3. **Formula for the quarter-circle arc length:** The circumference of a full circle is $2\pi r$. A quarter-circle arc length is $\frac{1}{4} \times 2\pi r = \frac{\pi r}{2}$. 4. **Calculate the quarter-circle arc length:** $$\text{Arc length} = \frac{\pi \times 4}{2} = 2\pi$$ 5. **Calculate the total length of wood:** The framework includes: - The quarter-circle arc length: $2\pi$ - The two straight edges along the radius (each 4 m), which form the quarter-circle boundary. - The length of the greenhouse (depth): 8 m Since the quarter-circle is the cross-section, the wood needed includes: - The quarter-circle arc (top curved part): $2\pi$ - The two straight radius edges (vertical and horizontal sides): $4 + 4 = 8$ - The length of the greenhouse (depth) along the straight edges: $8$ m 6. **Sum all parts:** $$\text{Total length} = 2\pi + 8 + 8 = 16 + 2\pi$$ 7. **Final answer:** The minimum total length of wood needed is $$16 + 2\pi$$ metres, where $a=16$ and $b=2$ are natural numbers.