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Consider the following subset of $\mathbb{R} ^{3}$ (the first two are cylinder, the third is a plane):

  • $C_{1}=\left \{ \left ( x,y,z \right ): y^{2}+z^{2}\leq 1 \right \};$
  • $C_{2}=\left \{ \left ( x,y,z \right ): x^{2}+z^{2}\leq 1 \right \};$
  • $H=\left \{ \left ( x,y,z \right ): z=0.2 \right \};$


Let $A = C_{1}\cap C_{2}\cap H.$ Which of the following best describe the shape of set $A?$

  1. Circle
  2. Ellipse
  3. Triangle
  4. Square
  5. An octagonal convex figure with curved sides
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The intersection of two perpendicular cylinders forms a solid called Steinmetz Solid. And the intersection with the plane $z=2$ can be visualized as below.

The enclosed intersection is a square of sides $a$, where $a^2=r^2+z^2$ (Apply Pythagoras Theorem). And here $a = \sqrt{1-z^2}$.


[[Image source: math.stackexchange.com]]

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