1. **Stating the problem:** We have three vase-like shapes with given surface areas, volumes, and heights. We want to analyze the first shape (A) which has a surface area of 50 cm² and a height of 5 cm. 2. **Formula and rules:** For 3D shapes like vases, surface area (SA) and volume (V) depend on the shape's geometry. Since the exact shape is not specified, we assume it is a cylinder-like shape for simplicity. - Surface area of a cylinder: $$SA = 2\pi r^2 + 2\pi r h$$ - Volume of a cylinder: $$V = \pi r^2 h$$ 3. **Given:** - Surface area $$SA = 50$$ cm² - Height $$h = 5$$ cm 4. **Find the radius $$r$$:** Using the surface area formula, $$50 = 2\pi r^2 + 2\pi r \times 5$$ 5. **Simplify:** $$50 = 2\pi r^2 + 10\pi r$$ 6. **Divide both sides by $$2\pi$$ to simplify:** $$\frac{50}{2\pi} = r^2 + 5r$$ Intermediate step with cancellation: $$\frac{\cancel{50}}{\cancel{2}\pi} = r^2 + 5r$$ Calculate $$\frac{50}{2\pi} = \frac{25}{\pi} \approx 7.96$$ 7. **Rewrite the quadratic equation:** $$r^2 + 5r - 7.96 = 0$$ 8. **Solve quadratic using the formula:** $$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=5$$, $$c=-7.96$$ Calculate discriminant: $$\Delta = 5^2 - 4 \times 1 \times (-7.96) = 25 + 31.84 = 56.84$$ Calculate roots: $$r = \frac{-5 \pm \sqrt{56.84}}{2} = \frac{-5 \pm 7.54}{2}$$ 9. **Select positive root:** $$r = \frac{-5 + 7.54}{2} = \frac{2.54}{2} = 1.27$$ cm 10. **Final answer:** The radius of the vase is approximately $$1.27$$ cm. This completes the solution for the first vase (A).