The shape L is extruded along Z-axis to form a solid as shown in the figure. How many shapes in the Set K will have more than 8 surfaces when extruded along Z-axis?
Show Hint
In extrusion problems, the number of side faces of the resulting 3D object is always equal to the number of edges of the original 2D shape. Carefully trace the perimeter of complex shapes to count the edges accurately. Remember the distinction between "greater than" ($>$) and "greater than or equal to" ($\geq$).
Step 1: Understanding the Question The problem asks us to determine how many of the 2D shapes in Set K will result in a 3D solid with more than 8 surfaces when extruded along the Z-axis. Step 2: Key Formula or Approach When a 2D polygon with \(n\) sides is extruded, it forms a 3D solid. The total number of surfaces of this solid is given by the formula: \[ \text{Total Surfaces} = 2 + n \] Here, '2' represents the front and back faces (which are the original 2D shape), and '\(n\)' represents the number of side surfaces, with one side surface corresponding to each edge of the original 2D shape. The condition is that the number of surfaces must be greater than 8: \[ 2 + n>8 \] \[ n>6 \] So, we need to count how many shapes in Set K have more than 6 sides (or edges). Step 3: Detailed Explanation Let's count the number of sides for each shape in Set K: 1. Plus sign (+): By tracing the perimeter, we can count the number of straight edges. It has 12 sides. - Surfaces = 2 + 12 = 14. Since 14 $>$ 8, this shape qualifies. 2. Shape J: Counting the straight edges around the perimeter gives 8 sides. - Surfaces = 2 + 8 = 10. Since 10 $>$ 8, this shape qualifies. 3. Shape V: This shape is a simple polygon with 6 sides. - Surfaces = 2 + 6 = 8. The condition is ""more than 8 surfaces"", so 8 does not qualify. 4. Shape K: Counting the sides along its perimeter gives 12 sides. - Surfaces = 2 + 12 = 14. Since 14 $>$ 8, this shape qualifies. 5. Semicircle with rectangle: This shape has 3 straight sides and 1 curved side. In the context of extrusion, each of these forms a surface. So, we can consider \(n=4\). - Surfaces = 2 (front/back) + 3 (flat sides) + 1 (curved side) = 6. Since 6 is not greater than 8, this shape does not qualify. Step 4: Final Answer The shapes that will have more than 8 surfaces when extruded are the plus sign (+), shape J, and shape K. There are a total of 3 such shapes. Therefore, the correct option is (B).
