What does cross-sections perpendicular mean?

In summary, The solid S has a circular base with a radius of two in the xy-plane, centered at the origin. The cross-sections of the solid perpendicular to the base are squares. The size of the squares depends on the angle of the knife and the x-axis, with the length of the sides being 2√(4-x2). The total area of the square is 4(4-x2).
  • #1
Allan
2
0
I am having trouble understanding a question to a
volume problem.

The base of a solid S is a circle of radius two in the
xy plane centered at the origin. Cross-sections of
the solid perpendicular to the base are squares.

I am thinking y = 4-X^2. Where do I find the
shape for the squares? Can someone explain?
 
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  • #2
The shape for the squares? Squares are squares!

I assume you mean the size of the squares.

To answer the title of you post, actually, saying "cross sections of the solid perpendicular to the base" is ambguous. The base is a circle of radius 2 centered at the origin (so x2+ y2 or y= +/-√(4- x2) (which is NOT quite what you have) . Imagine you have this object actually sitting in front of you. Take a sharp knife and slice through it. What you the cut side looks like (the cross section), depends on the angle the knife makes with the x and y axes as well as being perpendicular to the xy-plane.

I'm going to assume that cross-sections perpendicular to the x-axis are squares. (You hold your knife at right angles to the x-axis as you cut through the figure. The "cut end" looks like a square). A line through the figure, perpendicular to the x-axis runs from -√(4- x2) to +√(4- s2), a total length of 2&radic(4- x2). Being a square, the other sides are the same length and the area of the square is (2√(4-x2))2= 4(4-x2).
 
  • #3
Thanks

I can see it now
 

1. What is a cross-section?

A cross-section is a 2-dimensional representation of a 3-dimensional object or space, obtained by cutting through the object or space at a specific point or plane.

2. What does perpendicular mean in relation to cross-sections?

Perpendicular refers to two lines or surfaces that intersect at a 90 degree angle. In the context of cross-sections, it means that the cutting plane is perpendicular to the object or space.

3. How is a cross-section perpendicular to an object or space obtained?

A cross-section perpendicular to an object or space can be obtained by making a cut or plane that is perpendicular to the surface of the object or space at a specific point.

4. Why are cross-sections perpendicular to an object or space important?

Cross-sections perpendicular to an object or space provide a visual representation of the internal structure and features of the object or space. They are useful in understanding the shape, dimensions, and properties of the object or space.

5. Can cross-sections be non-perpendicular?

Yes, cross-sections can be non-perpendicular if the cutting plane is at an angle other than 90 degrees to the object or space. However, perpendicular cross-sections are more commonly used and provide a clearer representation of the object or space.

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