Around which line would the following cross-section need to be revolved to create a sphere?

Question

circle on a coordinate plane with center at 1 on the x-axis and a radius of 1

y = x
y-axis
x = 1
y = 1

Answer 1

To create a sphere by revolving a cross-section around a specific axis, we first need to visualize the given circle based on the provided information.

1. Circle Description: The circle is centered at (1, 0) on the x-axis with a radius of 1. This means the equation of the circle can be written as: \[ (x - 1)^2 + y^2 = 1 \] This circle will have points ranging from \( (0, 0) \) to \( (2, 0) \) along the x-axis and will reach a maximum y-value of 1 (at points (1, 1) and (1, -1)).

2. Identify Cross-section: The given lines involve:

   • \( y = x \): A line passing through the origin at a 45-degree angle.
   • The y-axis: where \( x = 0 \).
   • The vertical line \( x = 1 \): Indicates the line along the center of the circle.
   • The horizontal line \( y = 1 \): The upper limit of the circle.

3. Revolving to Form a Sphere: To create a sphere by revolving the circle around a line, we look for an axis of rotation. Notably:

   • We want the axis of rotation to be such that the entire circle is on one side of the line and is symmetric and evenly distributed about that line to maintain spherical shape.

4. Choosing the Line: The axis of rotation can be conveniently placed at the vertical line \( x = 1 \). This means we would revolve the figure (circle) around the line \( x = 1 \).

5. Conclusion: By revolving the circle defined by its center and radius around the line \( x = 1 \), we create a sphere.

Thus, the answer is: The circle should be revolved around the line \( x = 1 \).