Help me with this homework question.---**Question Number:** 18 **Question Stem:** A solid frustum is made by removing a small cone from a large cone as shown in the diagram. **Diagram Description (Frustum):** * Type: Geometric figure, specifically a frustum formed by cutting the top off a cone parallel to the base. * Main Elements: * Large cone: Represented by dashed lines for the removed top part. * Frustum: The solid shape shown below the cut. * Base: A large circular base at the bottom. * Top: A smaller circular top surface. * Slant height: A line segment along the sloping side. * Radius: A line segment from the center of the base to the edge. * Labels and Annotations: * "10 cm" label with a double-headed arrow indicating the slant height of the large cone. * "6 cm" label with a double-headed arrow indicating the slant height of the small (removed) cone. * "3 cm" label with a line segment indicating the radius of the base of the large cone. * Dashed lines represent the original large cone before the top was removed. **Textual Information:** The slant height of the small cone is 6 cm. The slant height of the large cone is 10 cm. The radius of the base of the large cone is 3 cm. **Calculation Task:** Calculate the total surface area of the frustum. **Formatting Requirement:** Give your answer correct to 3 significant figures. **Diagram Description (Cone Formula):** * Type: Geometric figure and formula box. * Main Elements: * Box containing text and a diagram. * Text "Curved surface area of cone = $\pi rl$". * Diagram of a cone showing the radius 'r' (from center of base to edge) and slant height 'l' (along the sloping side) labeled. **Mathematical Formula:** Curved surface area of cone = $\pi rl$

视频信息

答案文本 复制

视频字幕 复制

In this problem, we need to calculate the total surface area of a frustum. A frustum is formed by removing a small cone from a large cone. We're given that the slant height of the large cone is 10 centimeters, the slant height of the small cone is 6 centimeters, and the radius of the large cone's base is 3 centimeters. To find the total surface area, we'll need to find the area of the large circular base, the area of the small circular top, and the curved surface area of the frustum. We'll use the formula for the curved surface area of a cone, which is pi times radius times slant height. First, we need to find the radius of the small cone. Since the large and small cones are similar, the ratio of their corresponding sides is equal. This means the ratio of the small cone's radius to the large cone's radius equals the ratio of their slant heights. The ratio of slant heights is 6 to 10, which simplifies to 3 to 5. So, the ratio of the small radius to the large radius is also 3 to 5. Given that the large radius is 3 centimeters, we can write that r-sub-s divided by 3 equals 3 divided by 5. Solving for r-sub-s, we get 3 times 3 divided by 5, which equals 9 divided by 5, or 1.8 centimeters. Now that we know the radius of the small cone is 1.8 centimeters, we can calculate the areas of the circular bases and the curved surface. For the large circular base, the area equals pi times the square of the radius, which is pi times 3 squared, giving us 9 pi square centimeters. For the small circular top, the area equals pi times the square of the small radius, which is pi times 1.8 squared, giving us 3.24 pi square centimeters. For the curved surface area, we need to find the difference between the curved surface areas of the large and small cones. The curved surface area of the large cone is pi times radius times slant height, which is pi times 3 times 10, giving us 30 pi square centimeters. The curved surface area of the small cone is pi times 1.8 times 6, giving us 10.8 pi square centimeters. Therefore, the curved surface area of the frustum is 30 pi minus 10.8 pi, which equals 19.2 pi square centimeters. Now we can calculate the total surface area of the frustum by adding all three components: the area of the large base, the area of the small top, and the curved surface area. The total surface area equals 9 pi plus 3.24 pi plus 19.2 pi, which gives us 31.44 pi square centimeters. To find the numerical value, we multiply 31.44 by pi, which is approximately 3.14159, giving us about 98.78849 square centimeters. Rounding to three significant figures as required, our final answer is 98.8 square centimeters. Let's summarize what we've learned about calculating the surface area of a frustum. A frustum is formed by removing a small cone from a large cone. We used the similarity of triangles to find that the radius of the small cone is 1.8 centimeters. The total surface area consists of three components: the large circular base with area 9 pi square centimeters, the small circular top with area 3.24 pi square centimeters, and the curved surface area of 19.2 pi square centimeters. Adding these together, we get a total surface area of 31.44 pi, which is approximately 98.8 square centimeters when rounded to three significant figures. This problem demonstrates how to break down a complex shape into simpler components and use geometric principles to find the total surface area.