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Chapter 11: Problem 33

For some surfaces, the normal lines at any point pass through the same geometric object. What is the common geometric object for a sphere? What is the common geometric object for a right circular cylinder? Explain.

Short Answer

Expert verified
The common geometric object for a sphere is the center of the sphere. For a right circular cylinder, it is the line passing through the centers of the two bases.

Step by step solution

01

Understanding the shape and normal lines of a Sphere

A sphere is a three-dimensional geometric figure where all points on the surface are equidistant from the center. The normal line at any point on a sphere will point directly towards or away from the center of the sphere, as it is perpendicular to the tangent plane at that point. Any normal line drawn from the spherical surface will pass through the center of the sphere.
02

Finding the common geometric object for a Sphere

Since all normal lines of a sphere pass through its center, it is apparent that the common geometric object for the sphere is its center.
03

Understanding the shape and normal lines of a Right Circular Cylinder

A right circular cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting the bases. For any point on the curved surface of the right circular cylinder, the normal line will be parallel to the line connecting the centers of the bases.
04

Finding the common geometric object for a Right Circular Cylinder

For a right circular cylinder, the common geometric object is not a specific point, instead it is the entire line that passes through the centers of the circular bases. This is because the normal lines at any point on the curved surface are parallel to this line.

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Most popular questions from this chapter

The function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) \(f(x, y)=e^{x / y}\)

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