The coordinates A(0, 0), B(0, 3), C(4, 3) and D(4, 0) are graphed and connected together. What 3-dimensional shape will be created when this shape is rotated about the y-axis? Provide an explanation and proof for your answer to receive full credit.
My answer
A line segment is formed on the y-axis, formed by the points A(0,0) and B(0,3)
4 units away from the initial segment created by A, and B, is C(4,3) and D(4,0) and is parallel to the first line segment
A cylinder is formed when you rotate this around the y-axis, the cylinder has a radius of 4 units, and a height of 3 units.
To get the answer, we need to visualize the rotation of the given shape around the y-axis. In this case, we are rotating a 2-dimensional shape, which is a rectangle, around the y-axis.
First, let's analyze the initial shape formed by points A(0, 0), B(0, 3), C(4, 3), and D(4, 0). It is a rectangle with side AB measuring 3 units and side AD measuring 4 units. The rectangle is positioned in the xy-plane, with AB parallel to the y-axis and AD parallel to the x-axis.
Now, imagine rotating this rectangle around the y-axis. The y-axis acts as the axis of rotation. As the rectangle rotates, every point on it will remain at the same distance from the y-axis. Therefore, the shape formed by the rotation will be symmetrical on both sides of the y-axis.
The rotated shape will be a cylinder. The y-axis will be the central axis of the cylinder, and the rectangle formed by points A, B, C, and D will be the base of the cylinder.
To prove that the shape formed is indeed a cylinder, we can examine the properties of a cylinder and verify if they hold true for this rotated shape.
1. The base of the cylinder: The rectangle formed by points A, B, C, and D acts as the base of the cylinder.
2. The height of the cylinder: The height of the cylinder is equal to the distance between points B and C along the y-axis, which is 3 units.
3. The radius of the cylinder: The radius of the cylinder is the distance between the y-axis and any point on the rectangle. In this case, it is the distance between the y-axis and point C (4 units away from the y-axis).
By examining these properties, we can conclude that the shape formed by rotating the given rectangle around the y-axis is indeed a cylinder with a radius of 4 units and a height of 3 units.
To provide proof for this answer, we can visualize the 2-dimensional shape created by connecting the points A(0, 0), B(0, 3), C(4, 3), and D(4, 0). Drawing lines to connect these points forms a rectangle.
Now, consider rotating this 2-dimensional shape around the y-axis. The y-axis is a straight line passing through all points with an x-coordinate of 0. When we rotate the shape around the y-axis, each point maintains its distance from the y-axis but moves along a circular path.
For example, point A(0, 0) remains at a distance of 0 units from the y-axis as it rotates. Point B(0, 3) also remains at a distance of 3 units from the y-axis as it rotates. Similarly, points C(4, 3) and D(4, 0) maintain their distances from the y-axis during rotation.
By connecting these points as they rotate, we obtain a cylindrical shape. The distance between points C and D remains the height of the cylinder, which is 3 units in this case. The distance between the y-axis and points A or B remains the radius of the cylinder, which is 4 units in this case.
Therefore, when the shape created by connecting points A(0, 0), B(0, 3), C(4, 3), and D(4, 0) is rotated about the y-axis, a cylinder with a radius of 4 units and a height of 3 units is formed.
correct.
a rectangle is rotated about one of its sides.