Frustum Surface Area Aa494B

1. **State the problem:** We need to find the total surface area of a frustum of a cone with given dimensions: top diameter $12$ cm, bottom diameter $36$ cm, slant height $30$ cm, and vertical height $24$ cm. 2. **Recall the formula:** The curved surface area (CSA) of a cone is given by $$\text{CSA} = \pi r l$$ where $r$ is the radius and $l$ is the slant height. 3. **Frustum surface area:** The total surface area of a frustum includes: - Curved surface area of the frustum - Area of the top circular face - Area of the bottom circular face 4. **Calculate radii:** - Top radius $r_1 = \frac{12}{2} = 6$ cm - Bottom radius $r_2 = \frac{36}{2} = 18$ cm 5. **Curved surface area of frustum:** $$\text{CSA}_{frustum} = \pi (r_1 + r_2) l = \pi (6 + 18) \times 30 = \pi \times 24 \times 30$$ 6. **Calculate:** $$\text{CSA}_{frustum} = 720\pi$$ 7. **Area of top circle:** $$A_{top} = \pi r_1^2 = \pi \times 6^2 = 36\pi$$ 8. **Area of bottom circle:** $$A_{bottom} = \pi r_2^2 = \pi \times 18^2 = 324\pi$$ 9. **Total surface area:** $$\text{Total} = \text{CSA}_{frustum} + A_{top} + A_{bottom} = 720\pi + 36\pi + 324\pi = (720 + 36 + 324)\pi = 1080\pi$$ 10. **Numerical value:** $$1080\pi \approx 1080 \times 3.1416 = 3392.92$$ 11. **Final answer to 3 significant figures:** $$\boxed{3390 \text{ cm}^2}$$ This is the total surface area of the frustum.