Question

charlotte - mecklenburg schools math 3, unit 3, lesson 1 student are you ready for more? 1. graph y=-x + 3 from x = 1 to x = 3. 2. sketch the solid of rotation generated by rotating this line using the y - axis as the axis of rotation. 3. what figure is made? 4. the object being rotated here is a line and not a two - dimensional object like in the lesson. how does this affect the result of the rotation?

Explanation:

Step1: Find points on the line

When \(x = 1\), \(y=-1 + 3=2\); when \(x = 3\), \(y=-3 + 3=0\). Plot the points \((1,2)\) and \((3,0)\) and draw the line segment \(y=-x + 3\) for \(1\leq x\leq3\) on the graph.

Step2: Sketch the solid of rotation

Visualize rotating the line segment around the \(y\)-axis. The line segment will sweep out a frustum - like shape.

Step3: Identify the figure

The figure made is a frustum of a cone. A frustum of a cone is formed when a right - circular cone is cut by a plane parallel to the base of the cone. Here, rotating the line segment around the \(y\)-axis creates a shape that has two circular bases of different radii and a slanted surface connecting them.

Step4: Analyze the effect of rotating a line

When rotating a two - dimensional object, we get a solid with volume and surface area. When rotating a line, we get a surface (the lateral surface of the frustum in this case) and the two circular bases are formed by the endpoints of the line segment's rotation. The result is a surface - bounded solid rather than a solid that is 'filled in' in all directions from a 2D shape.

Answer:

1. Plot points \((1,2)\) and \((3,0)\) and draw the line segment \(y=-x + 3\) for \(1\leq x\leq3\).
2. Sketch a frustum - like shape around the \(y\)-axis.
3. Frustum of a cone.
4. Rotating a line gives a surface - bounded solid with the lateral surface formed by the line's rotation and circular bases from the endpoints' rotation, different from rotating a 2D object which gives a more 'filled - in' solid.