Math Insight

Applet: Ice cream cone region with shadow

Applet: Ice cream cone region with shadow

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The ice cream cone region is bounded above by the half-sphere $z=\sqrt{1-x^2-y^2}$ and bounded below by the cone $z=\sqrt{x^2+y^2}$. The two surfaces intersect along a circle defined by $x^2+y^2=1/2$ and $z=1/\sqrt{2}$, which is the widest part of the ice cream cone. Therefore, the shadow of the ice cream cone region parallel to the $xy$-plane is the disk of radius $1/\sqrt{2}$ described by $x^2+y^2 \le 1/2$.

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