1. **Stating the problem:** We have a large cone with height 45 cm and radius 9 cm. A smaller cone with height 4 cm is removed from the top, forming a frustum. We need to find the length $x$ (the radius of the smaller cone) and then calculate the total surface area of the frustum in terms of $\pi$. 2. **Finding the length $x$:** Since the small cone is similar to the large cone, their corresponding dimensions are proportional. The ratio of heights equals the ratio of radii: $$\frac{x}{9} = \frac{4}{45}$$ 3. **Solving for $x$:** $$x = 9 \times \frac{4}{45} = \frac{36}{45} = \frac{4}{5} = 0.8 \text{ cm}$$ 4. **Calculating the slant heights:** - Large cone slant height $l_1$: $$l_1 = \sqrt{9^2 + 45^2} = \sqrt{81 + 2025} = \sqrt{2106}$$ - Small cone slant height $l_2$: $$l_2 = \sqrt{0.8^2 + 4^2} = \sqrt{0.64 + 16} = \sqrt{16.64}$$ 5. **Calculating the frustum slant height $l$:** $$l = l_1 - l_2 = \sqrt{2106} - \sqrt{16.64}$$ 6. **Surface area of the frustum:** The total surface area of a frustum is the sum of the curved surface areas of the large and small cones minus the small cone's curved surface area (which is removed), plus the area of the top and bottom circles: Curved surface area of frustum: $$\text{CSA} = \pi (r_1 + r_2) l$$ where $r_1 = 9$, $r_2 = 0.8$, and $l$ is the frustum slant height. 7. **Calculate $l$ numerically:** $$l = \sqrt{2106} - \sqrt{16.64} \approx 45.9 - 4.08 = 41.82 \text{ cm}$$ 8. **Calculate curved surface area:** $$\text{CSA} = \pi (9 + 0.8)(41.82) = \pi (9.8)(41.82) = 409.84\pi$$ 9. **Calculate areas of top and bottom circles:** - Bottom circle area: $$\pi (9)^2 = 81\pi$$ - Top circle area: $$\pi (0.8)^2 = 0.64\pi$$ 10. **Total surface area of frustum:** $$\text{Total SA} = \text{CSA} + \text{top area} + \text{bottom area} = 409.84\pi + 0.64\pi + 81\pi = (409.84 + 0.64 + 81)\pi = 491.48\pi$$ **Final answers:** - Length $x = 0.8$ cm - Total surface area of the frustum $= 491.48\pi$ cm$^2$