consider the following figure: figure of a rectangle with length 3, height 2 on top and bottom, and a dashed line m through the middle what solid 3d object is produced by rotating the rectangle about line m? choose 1 answer: a a cylinder with diameter 3 units b a cylinder with diameter 4 units c a cone with diameter 3 units d a cone with diameter 4 units

Question

Accepted Answer

# Brief Explanations:
When a rectangle is rotated about a line parallel to one of its sides (here, line $ m $ is parallel to the side of length 3 and the distance from the line to the vertical sides is 2, so the radius of the cylinder formed will be 2, making the diameter $ 2\times2 = 4 $? Wait, no, wait. Wait, the rectangle has a vertical side? Wait, no, the rectangle has length 3 (horizontal) and the vertical sides: the distance from line $ m $ to each vertical side? Wait, no, the rectangle is rotated about line $ m $. The rectangle has a side of length 3 (horizontal) and the vertical dimension: from the top 2 to bottom 2, so the total height? Wait, no, when you rotate a rectangle about a line parallel to one of its sides, you get a cylinder. The radius of the cylinder is the distance from the line of rotation to the side perpendicular to the line. Here, the line $ m $ is parallel to the horizontal side (length 3) and the vertical sides are at a distance of 2 from the line? Wait, no, the rectangle has a horizontal side of length 3, and the vertical sides: the distance from line $ m $ to the left and right? Wait, no, the rectangle is symmetric about line $ m $. The vertical sides: the distance from line $ m $ to each vertical side? Wait, the rectangle has a height (vertical) of $ 2 + 2 = 4 $? No, wait, the figure shows a rectangle with top 2, bottom 2, and horizontal length 3. So when rotating about line $ m $ (the dashed line), which is parallel to the horizontal side (length 3) and passes through the center of the vertical sides. So the side of the rectangle perpendicular to line $ m $ is the vertical side? Wait, no, line $ m $ is horizontal? Wait, the arrows are horizontal, so line $ m $ is horizontal. The rectangle has a horizontal length of 3 and vertical height of $ 2 + 2 = 4 $? No, the rectangle is a square? No, it's a rectangle with length 3 (horizontal) and height 4 (vertical, 2 + 2). Wait, no, the figure shows a rectangle with top 2, bottom 2, so the total vertical length is $ 2 + 2 = 4 $, and horizontal length 3. When rotating about the horizontal line $ m $, the side perpendicular to line $ m $ is the vertical side (length 3). Wait, no, rotation about a horizontal line: the rectangle is rotated around line $ m $, so the side parallel to line $ m $ is the horizontal side (length 3), and the side perpendicular to line $ m $ is the vertical side (length $ 2 + 2 = 4 $)? No, that can't be. Wait, no, the rectangle has a horizontal side of length 3, and the vertical sides: the distance from line $ m $ to each vertical side is 2? Wait, no, the rectangle is symmetric about line $ m $, so the distance from line $ m $ to the left and right vertical sides? Wait, no, the rectangle is a vertical rectangle? Wait, the figure shows a rectangle with right angles, top and bottom 2, and horizontal length 3. So when rotating about line $ m $ (horizontal), the rectangle will form a cylinder. The radius of the cylinder is the distance from line $ m $ to the side perpendicular to line $ m $. Wait, line $ m $ is horizontal, so the sides perpendicular to line $ m $ are vertical. The distance from line $ m $ to each vertical side is 2? No, the rectangle has a horizontal length of 3, so the distance from line $ m $ to the left and right vertical sides: since the rectangle is symmetric, the distance from line $ m $ to the left vertical side is $ 3/2 $? No, that can't be. Wait, maybe I got the axes wrong. Let's recall: when you rotate a rectangle about a line parallel to one of its sides, you get a cylinder. The radius of the cylinder is the length of the side perpendicular to the axis of rotation, and the height is the length of the side parallel to the axis. Wait, no: if you rotate a rectangle with length $ l $ and width $ w $ about the line parallel to the length (so the axis is along the length), then the radius is $ w $ and the height is $ l $. Wait, in this case, the axis of rotation (line $ m $) is parallel to the horizontal side (length 3) of the rectangle. So the side of the rectangle perpendicular to the axis is the vertical side. Wait, the vertical side: the distance from the axis to the vertical side? Wait, the rectangle has a vertical dimension of $ 2 + 2 = 4 $? No, the figure shows a rectangle with top 2, bottom 2, so the total vertical length is $ 2 + 2 = 4 $, and horizontal length 3. So when rotating about the horizontal axis (line $ m $), the side perpendicular to the axis is the horizontal side? No, I'm confused. Wait, let's think of the rectangle: it's a rectangle with length 3 (horizontal) and height 4 (vertical, 2 + 2). The axis of rotation (line $ m $) is horizontal, passing through the center of the vertical sides. So when we rotate the rectangle about line $ m $, the horizontal side (length 3) will be the height of the cylinder, and the vertical side (distance from line $ m $ to the top and bottom) is the radius? Wait, no, the top and bottom are each 2 units from line $ m $, so the radius is 2, so the diameter is $ 2 \times 2 = 4 $. And the height of the cylinder is the horizontal length of the rectangle, which is 3. So the solid formed is a cylinder with diameter 4 units? Wait, no, the options: option A is diameter 3, option B is diameter 4. Wait, maybe I mixed up the sides. Wait, the rectangle has a horizontal length of 3 and vertical height of 4 (2 + 2). When rotating about line $ m $ (horizontal), the side perpendicular to the axis is the vertical side? No, the axis is horizontal, so the sides parallel to the axis are horizontal, and the sides perpendicular to the axis are vertical. So the length of the side perpendicular to the axis is the vertical length, which is 4? No, that can't be. Wait, no, the rectangle is a rectangle with length 3 (horizontal) and width 2 (vertical)? Wait, the figure shows a rectangle with top 2, bottom 2, so maybe the vertical side is 2, and the horizontal side is 3, and the line $ m $ is parallel to the horizontal side, at a distance of 2 from the top and bottom? No, that doesn't make sense. Wait, maybe the rectangle has a height of 2 (top) and 2 (bottom), so total height 4, and width 3 (horizontal). When rotating about line $ m $ (the horizontal dashed line), the radius of the cylinder is the distance from line $ m $ to the top or bottom, which is 2, so diameter is $ 2 \times 2 = 4 $, and the height of the cylinder is the width of the rectangle, which is 3. So the solid is a cylinder with diameter 4 units. So option B.
# Answer:
B. A cylinder with diameter 4 units

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: