1. **State the problem:** We have a quarter of a solid sphere (Shape S) with volume $576\pi$ cm³. We need to find the surface area of this quarter sphere, rounded to 3 significant figures. 2. **Recall formulas:** - Volume of a sphere: $$V = \frac{4}{3} \pi r^3$$ - Surface area of a sphere: $$A = 4 \pi r^2$$ 3. **Find the radius $r$ of the full sphere:** Since Shape S is one quarter of the sphere, its volume is: $$V_S = \frac{1}{4} \times \frac{4}{3} \pi r^3 = \frac{1}{3} \pi r^3$$ Given $V_S = 576\pi$, set equal: $$\frac{1}{3} \pi r^3 = 576 \pi$$ Divide both sides by $\pi$: $$\frac{1}{3} r^3 = 576$$ Multiply both sides by 3: $$r^3 = 1728$$ Take cube root: $$r = \sqrt[3]{1728} = 12 \text{ cm}$$ 4. **Calculate the surface area of the quarter sphere:** The quarter sphere surface area includes: - One quarter of the curved surface area of the full sphere - Plus the areas of the two flat quarter-circle faces (the two flat surfaces formed by cutting the sphere into quarters) Curved surface area of full sphere: $$4 \pi r^2 = 4 \pi (12)^2 = 4 \pi \times 144 = 576 \pi$$ Quarter curved surface area: $$\frac{1}{4} \times 576 \pi = 144 \pi$$ Each flat face is a quarter circle of radius $r$, so area of one flat face: $$\frac{1}{4} \pi r^2 = \frac{1}{4} \pi (12)^2 = 36 \pi$$ There are two such flat faces, so total flat area: $$2 \times 36 \pi = 72 \pi$$ 5. **Total surface area of Shape S:** $$144 \pi + 72 \pi = 216 \pi$$ 6. **Calculate numerical value:** $$216 \pi \approx 216 \times 3.1416 = 678.584$$ Rounded to 3 significant figures: $$\boxed{679 \text{ cm}^2}$$