Question

each region below is being rotated about the line l. for each region, identify the solid generated by the rotation. region (a) (b) (c) solid

Explanation:

Part (a)

Step 1: Analyze the Region

The region in (a) is a trapezoid (a quadrilateral with one pair of parallel sides) rotated about the vertical line \( l \). The bottom part is a rectangle (when rotated forms a cylinder) and the top part is a right triangle (when rotated forms a cone). So the solid should be a combination of a cylinder and a cone (a cylinder with a conical top).

Step 2: Identify the Solid

Looking at the options, the second solid (from the left) has a cylindrical base and a conical top, which matches the rotation of the trapezoid. So for (a), the correct solid is the second one (the one with the cylinder and cone on top).

Part (b)

Step 1: Analyze the Region

The region in (b) is a triangle (a right triangle) rotated about the vertical line \( l \). Wait, no, the region is a triangle? Wait, no, looking at the figure, it's a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? 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Answer:

Part (a)

Step 1: Analyze the Region

The region in (a) is a trapezoid (a quadrilateral with one pair of parallel sides) rotated about the vertical line \( l \). The bottom part is a rectangle (when rotated forms a cylinder) and the top part is a right triangle (when rotated forms a cone). So the solid should be a combination of a cylinder and a cone (a cylinder with a conical top).

Step 2: Identify the Solid

Looking at the options, the second solid (from the left) has a cylindrical base and a conical top, which matches the rotation of the trapezoid. So for (a), the correct solid is the second one (the one with the cylinder and cone on top).

Part (b)

Step 1: Analyze the Region

The region in (b) is a triangle (a right triangle) rotated about the vertical line \( l \). Wait, no, the region is a triangle? Wait, no, looking at the figure, it's a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? Wait, no, the region is a triangle? 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